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<title>Implicit function</title>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation>

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</math></span><img src="./17d063dc86a53a2efb1fe86f4a5d47d498652766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}" loading="lazy"></span></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;">

<ul><li><a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul>

<ul><li><a href="Limit_of_a_function" title="Limit of a function">Limits</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuity</a></li></ul>

</div><div class="hlist">

<ul><li><a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a></li>

<li><a href="Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li>

<li><a href="Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul>

</div></td></tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><span style="font-size:120%"><a href="Differential_calculus" title="Differential calculus">Differential</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">

Definitions</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Derivative" title="Derivative">Derivative</a> (<a href="Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li>

<li><a href="Differential_(mathematics)" title="Differential (mathematics)">Differential</a>

<ul><li><a href="Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li>

<li><a href="Differential_of_a_function" title="Differential of a function">of a function</a></li>

<li><a href="Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td>

</tr><tr><th class="sidebar-heading">

Concepts</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li>

<li><a href="Second_derivative" title="Second derivative">Second derivative</a></li>

<li><a href="Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li>

<li><a href="Related_rates" title="Related rates">Related rates</a></li>

<li><a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

<a href="Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist">

<li><a href="Product_rule" title="Product rule">Product</a></li>

<li><a href="Chain_rule" title="Chain rule">Chain</a></li>

<li><a href="Power_rule" title="Power rule">Power</a></li>

<li><a href="Quotient_rule" title="Quotient rule">Quotient</a></li>

<li><a href="L'H%C3%B4pital's_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li>

<li><a href="Inverse_function_rule" title="Inverse function rule">Inverse</a></li>

<li><a href="General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li>

<li><a href="Fa%C3%A0_di_Bruno's_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li>

<li><a href="Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td>

</tr></tbody></table></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Integral" title="Integral">Integral</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transform</a></li>

<li><a href="Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Definitions</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Antiderivative" title="Antiderivative">Antiderivative</a></li>

<li><a href="Integral" title="Integral">Integral</a> (<a href="Improper_integral" title="Improper integral">improper</a>)</li>

<li><a href="Riemann_integral" title="Riemann integral">Riemann integral</a></li>

<li><a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li>

<li><a href="Contour_integration" title="Contour integration">Contour integration</a></li>

<li><a href="Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Integration by</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Integration_by_parts" title="Integration by parts">Parts</a></li>

<li><a href="Disc_integration" title="Disc integration">Discs</a></li>

<li><a href="Shell_integration" title="Shell integration">Cylindrical shells</a></li>

<li><a href="Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="Euler_substitution" title="Euler substitution">Euler</a>)</li>

<li><a href="Integration_using_Euler's_formula" title="Integration using Euler's formula">Euler's formula</a></li>

<li><a href="Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li>

<li><a href="Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li>

<li><a href="Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li>

<li><a href="Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li>

<li><a href="Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Series_(mathematics)" title="Series (mathematics)">Series</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Geometric_series" title="Geometric series">Geometric</a> (<a href="Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li>

<li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Power_series" title="Power series">Power</a></li>

<li><a href="Binomial_series" title="Binomial series">Binomial</a></li>

<li><a href="Taylor_series" title="Taylor series">Taylor</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

<a href="Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li>

<li><a href="Ratio_test" title="Ratio test">Ratio</a></li>

<li><a href="Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li>

<li><a href="Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li>

<li><br><a href="Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li>

<li><a href="Alternating_series_test" title="Alternating series test">Alternating series</a></li>

<li><a href="Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li>

<li><a href="Dirichlet's_test" title="Dirichlet's test">Dirichlet</a></li>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Vector_calculus" title="Vector calculus">Vector</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Gradient" title="Gradient">Gradient</a></li>

<li><a href="Divergence" title="Divergence">Divergence</a></li>

<li><a href="Laplace_operator" title="Laplace operator">Laplacian</a></li>

<li><a href="Directional_derivative" title="Directional derivative">Directional derivative</a></li>

<li><a href="Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Theorems</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Gradient_theorem" title="Gradient theorem">Gradient</a></li>

<li><a href="Green's_theorem" title="Green's theorem">Green's</a></li>

<li><a href="Stokes'_theorem" title="Stokes' theorem">Stokes'</a></li>

<li><a href="Divergence_theorem" title="Divergence theorem">Divergence</a></li>

<li><a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li>

<li><a href="Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">

Formalisms</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Matrix_calculus" title="Matrix calculus">Matrix</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior</a></li>

<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Definitions</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Multiple_integral" title="Multiple integral">Multiple integral</a></li>

<li><a href="Line_integral" title="Line integral">Line integral</a></li>

<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li>

<li><a href="Volume_integral" title="Volume integral">Volume integral</a></li>

<li><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li>

<li><a href="Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Advanced</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li>

<li><a href="Generalized_function" title="Generalized function">Generalized functions</a></li>

<li><a href="Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Specialized</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;">

<ul><li><a href="Fractional_calculus" title="Fractional calculus">Fractional</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li>

<li><a href="Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Miscellanea</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;">

<ul><li><a href="Precalculus" title="Precalculus">Precalculus</a></li>

<li><a href="History_of_calculus" title="History of calculus">History</a></li>

<li><a href="Glossary_of_calculus" title="Glossary of calculus">Glossary</a></li>

<li><a href="List_of_calculus_topics" title="List of calculus topics">List of topics</a></li>

<li><a href="Integration_Bee" title="Integration Bee">Integration Bee</a></li>

<li><a href="Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li>

<li><a href="Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>implicit equation</b> is a <a href="Relation_(mathematics)" title="Relation (mathematics)">relation</a> of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x_{1},\dots ,x_{n})=0,}">

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<annotation encoding="application/x-tex">{\displaystyle R(x_{1},\dots ,x_{n})=0,}</annotation>

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</math></span><img src="./65fe185e4df1af148fe40681cc65538e310efa4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.591ex; height:2.843ex;" alt="{\displaystyle R(x_{1},\dots ,x_{n})=0,}" loading="lazy"></span> where <span class="texhtml mvar" style="font-style:italic;">R</span> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of several variables (often a <a href="Polynomial" title="Polynomial">polynomial</a>). For example, the implicit equation of the <a href="Unit_circle" title="Unit circle">unit circle</a> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}-1=0.}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}-1=0.}</annotation>

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</math></span><img src="./6b118fda6a2c7179d1e0589937147bfbb25625fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.35ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-1=0.}" loading="lazy"></span>

</p><p>An <b>implicit function</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that is defined by an implicit equation, that relates one of the variables, considered as the <a href="Value_(mathematics)" title="Value (mathematics)">value</a> of the function, with the others considered as the <a href="Argument_of_a_function" title="Argument of a function">arguments</a>.<sup id="cite_ref-Chiang_1-0" class="reference"><a href="#cite_note-Chiang-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 204–206">: 204–206 </span></sup> For example, the equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}-1=0}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}-1=0}</annotation>

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</math></span><img src="./1ee594b8851d760d0e2d44aba714907aca657b8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.703ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-1=0}" loading="lazy"></span> of the <a href="Unit_circle" title="Unit circle">unit circle</a> defines <span class="texhtml mvar" style="font-style:italic;">y</span> as an implicit function of <span class="texhtml mvar" style="font-style:italic;">x</span> if <span class="texhtml">−1 ≤ <i>x</i> ≤ 1</span>, and <span class="texhtml mvar" style="font-style:italic;">y</span> is restricted to nonnegative values.

</p><p>The <a href="Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a> provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero <a href="Multivariable_function" class="mw-redirect" title="Multivariable function">multivariable functions</a> that are <a href="Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Inverse_functions">Inverse functions</h3></div>

<p>A common type of implicit function is an <a href="Inverse_function" title="Inverse function">inverse function</a>. Not all functions have a unique inverse function. If <span class="texhtml mvar" style="font-style:italic;">g</span> is a function of <span class="texhtml mvar" style="font-style:italic;">x</span> that has a unique inverse, then the inverse function of <span class="texhtml mvar" style="font-style:italic;">g</span>, called <span class="texhtml"><i>g</i><sup>−1</sup></span>, is the unique function giving a <a href="Solution_(mathematics)" class="mw-redirect" title="Solution (mathematics)">solution</a> of the equation

</p>

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<annotation encoding="application/x-tex">{\displaystyle y=g(x)}</annotation>

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</math></span><img src="./26c08f8fd3471dad5e2c45c2f753ffd7c9aba4ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.509ex; height:2.843ex;" alt="{\displaystyle y=g(x)}" loading="lazy"></span></dd></dl>

<p>for <span class="texhtml mvar" style="font-style:italic;">x</span> in terms of <span class="texhtml mvar" style="font-style:italic;">y</span>. This solution can then be written as

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<annotation encoding="application/x-tex">{\displaystyle x=g^{-1}(y)\,.}</annotation>

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</math></span><img src="./31764a78c944f1c29301c78149f1c10d889dc0bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.878ex; height:3.176ex;" alt="{\displaystyle x=g^{-1}(y)\,.}" loading="lazy"></span></dd></dl>

<p>Defining <span class="texhtml"><i>g</i><sup>−1</sup></span> as the inverse of <span class="texhtml mvar" style="font-style:italic;">g</span> is an implicit definition. For some functions <span class="texhtml mvar" style="font-style:italic;">g</span>, <span class="texhtml"><i>g</i><sup>−1</sup>(<i>y</i>)</span> can be written out explicitly as a <a href="Closed-form_expression" title="Closed-form expression">closed-form expression</a> — for instance, if <span class="texhtml"><i>g</i>(<i>x</i>) = 2<i>x</i> − 1</span>, then <span class="texhtml"><i>g</i><sup>−1</sup>(<i>y</i>) = <style data-mw-deduplicate="TemplateStyles:r1214402035">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}

/* end https://en.wikipedia.org/ */

</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>(<i>y</i> + 1)</span>. However, this is often not possible, or only by introducing a new notation (as in the <a href="Product_log" class="mw-redirect" title="Product log">product log</a> example below).

</p><p>Intuitively, an inverse function is obtained from <span class="texhtml mvar" style="font-style:italic;">g</span> by interchanging the roles of the dependent and independent variables.

</p><p><b>Example:</b> The <a href="Product_log" class="mw-redirect" title="Product log">product log</a> is an implicit function giving the solution for <span class="texhtml mvar" style="font-style:italic;">x</span> of the equation <span class="texhtml"><i>y</i> − <i>xe</i><sup><i>x</i></sup> = 0</span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_functions">Algebraic functions</h3></div>

/* start https://en.wikipedia.org/ */

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Algebraic_function" title="Algebraic function">Algebraic function</a></div>

<p>An <b>algebraic function</b> is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable <span class="texhtml mvar" style="font-style:italic;">x</span> gives a solution for <span class="texhtml mvar" style="font-style:italic;">y</span> of an equation

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<annotation encoding="application/x-tex">{\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0\,,}</annotation>

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</math></span><img src="./4023ce7098805c4f2514b9233d6122e4db4431d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.096ex; height:3.176ex;" alt="{\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0\,,}" loading="lazy"></span></dd></dl>

<p>where the coefficients <span class="texhtml"><i>a<sub>i</sub></i>(<i>x</i>)</span> are polynomial functions of <span class="texhtml mvar" style="font-style:italic;">x</span>. This algebraic function can be written as the right side of the solution equation <span class="texhtml"><i>y</i> = <i>f</i>(<i>x</i>)</span>. Written like this, <span class="texhtml mvar" style="font-style:italic;">f</span> is a <a href="Multi-valued_function" class="mw-redirect" title="Multi-valued function">multi-valued</a> implicit function.

</p><p>Algebraic functions play an important role in <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> and <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. A simple example of an algebraic function is given by the left side of the unit circle equation:

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}-1=0\,.}</annotation>

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</math></span><img src="./813d43da03f1fad7399d308c579f44404954158e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.737ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-1=0\,.}" loading="lazy"></span></dd></dl>

<p>Solving for <span class="texhtml mvar" style="font-style:italic;">y</span> gives an explicit solution:

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<annotation encoding="application/x-tex">{\displaystyle y=\pm {\sqrt {1-x^{2}}}\,.}</annotation>

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</math></span><img src="./46efec109780a439c2572bc8d3a57b8872a6dc7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.807ex; height:3.509ex;" alt="{\displaystyle y=\pm {\sqrt {1-x^{2}}}\,.}" loading="lazy"></span></dd></dl>

<p>But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as <span class="texhtml"><i>y</i> = <i>f</i>(<i>x</i>)</span>, where <span class="texhtml mvar" style="font-style:italic;">f</span> is the multi-valued implicit function.

</p><p>While explicit solutions can be found for equations that are <a href="Quadratic_equations" class="mw-redirect" title="Quadratic equations">quadratic</a>, <a href="Cubic_equation" title="Cubic equation">cubic</a>, and <a href="Quartic_equation" title="Quartic equation">quartic</a> in <span class="texhtml mvar" style="font-style:italic;">y</span>, the same is not in general true for <a href="Quintic_equation" class="mw-redirect" title="Quintic equation">quintic</a> and higher degree equations, such as

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<annotation encoding="application/x-tex">{\displaystyle y^{5}+2y^{4}-7y^{3}+3y^{2}-6y-x=0\,.}</annotation>

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</math></span><img src="./b2179be275419315a0658bb00b47f21f2fc1ff67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.491ex; height:3.009ex;" alt="{\displaystyle y^{5}+2y^{4}-7y^{3}+3y^{2}-6y-x=0\,.}" loading="lazy"></span></dd></dl>

<p>Nevertheless, one can still refer to the implicit solution <span class="texhtml"><i>y</i> = <i>f</i>(<i>x</i>)</span> involving the multi-valued implicit function <span class="texhtml mvar" style="font-style:italic;">f</span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Caveats">Caveats</h2></div>

<p>Not every equation <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>) = 0</span> implies a graph of a single-valued function, the circle equation being one prominent example. Another example is an implicit function given by <span class="texhtml"><i>x</i> − <i>C</i>(<i>y</i>) = 0</span> where <span class="texhtml mvar" style="font-style:italic;">C</span> is a <a href="Cubic_polynomial" class="mw-redirect" title="Cubic polynomial">cubic polynomial</a> having a "hump" in its graph. Thus, for an implicit function to be a <i>true</i> (single-valued) function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a true function only after "zooming in" on some part of the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis and "cutting away" some unwanted function branches. Then an equation expressing <span class="texhtml mvar" style="font-style:italic;">y</span> as an implicit function of the other variables can be written.

</p><p>The defining equation <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>) = 0</span> can also have other pathologies. For example, the equation <span class="texhtml"><i>x</i> = 0</span> does not imply a function <span class="texhtml"><i>f</i>(<i>x</i>)</span> giving solutions for <span class="texhtml mvar" style="font-style:italic;">y</span> at all; it is a vertical line. In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the <a href="Function_domain" class="mw-redirect" title="Function domain">domain</a>. The <a href="Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a> provides a uniform way of handling these sorts of pathologies.

</p>

<div class="mw-heading mw-heading2"><h2 id="Implicit_differentiation">Implicit differentiation</h2></div>

<p>In <a href="Calculus" title="Calculus">calculus</a>, a method called <b>implicit differentiation</b> makes use of the <a href="Chain_rule" title="Chain rule">chain rule</a> to differentiate implicitly defined functions.

</p><p>To differentiate an implicit function <span class="texhtml"><i>y</i>(<i>x</i>)</span>, defined by an equation <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>) = 0</span>, it is not generally possible to solve it explicitly for <span class="texhtml mvar" style="font-style:italic;">y</span> and then differentiate. Instead, one can <a href="Total_differentiation" class="mw-redirect" title="Total differentiation">totally differentiate</a> <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>) = 0</span> with respect to <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> and then solve the resulting linear equation for <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dy</i></span><span class="sr-only">/</span><span class="den"><i>dx</i></span></span>⁠</span></span> to explicitly get the derivative in terms of <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>. Even when it is possible to explicitly solve the original equation, the formula resulting from total differentiation is, in general, much simpler and easier to use.

</p>

<div class="mw-heading mw-heading3"><h3 id="Examples_2">Examples</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Example_1">Example 1</h4></div>

<p>Consider

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<annotation encoding="application/x-tex">{\displaystyle y+x+5=0\,.}</annotation>

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</math></span><img src="./7062779ee7b9b549b4b00ce995cd0675a35b668c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.623ex; height:2.509ex;" alt="{\displaystyle y+x+5=0\,.}" loading="lazy"></span></dd></dl>

<p>This equation is easy to solve for <span class="texhtml mvar" style="font-style:italic;">y</span>, giving

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<annotation encoding="application/x-tex">{\displaystyle y=-x-5\,,}</annotation>

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</math></span><img src="./1880575102e8ad405c08390b966a105d979293ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.429ex; height:2.509ex;" alt="{\displaystyle y=-x-5\,,}" loading="lazy"></span></dd></dl>

<p>where the right side is the explicit form of the function <span class="texhtml"><i>y</i>(<i>x</i>)</span>. Differentiation then gives <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dy</i></span><span class="sr-only">/</span><span class="den"><i>dx</i></span></span>⁠</span> = −1</span>.

</p><p>Alternatively, one can totally differentiate the original equation:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {dy}{dx}}+{\frac {dx}{dx}}+{\frac {d}{dx}}(5)&=0\,;\\[6px]{\frac {dy}{dx}}+1+0&=0\,.\end{aligned}}}</annotation>

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</math></span><img src="./bbc6ffe0eb05941c2a2c5c5a76fcfcfe3f07311c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:24.844ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {dy}{dx}}+{\frac {dx}{dx}}+{\frac {d}{dx}}(5)&=0\,;\\[6px]{\frac {dy}{dx}}+1+0&=0\,.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Solving for <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dy</i></span><span class="sr-only">/</span><span class="den"><i>dx</i></span></span>⁠</span></span> gives

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}=-1\,,}</annotation>

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<p>the same answer as obtained previously.

</p>

<div class="mw-heading mw-heading4"><h4 id="Example_2">Example 2</h4></div>

<p>An example of an implicit function for which implicit differentiation is easier than using explicit differentiation is the function <span class="texhtml"><i>y</i>(<i>x</i>)</span> defined by the equation

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<annotation encoding="application/x-tex">{\displaystyle x^{4}+2y^{2}=8\,.}</annotation>

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</math></span><img src="./861fbac28deaf6ba918a3fd071e8700a2615f122.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.896ex; height:3.009ex;" alt="{\displaystyle x^{4}+2y^{2}=8\,.}" loading="lazy"></span></dd></dl>

<p>To differentiate this explicitly with respect to <span class="texhtml mvar" style="font-style:italic;">x</span>, one has first to get

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<annotation encoding="application/x-tex">{\displaystyle y(x)=\pm {\sqrt {\frac {8-x^{4}}{2}}}\,,}</annotation>

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<p>and then differentiate this function. This creates two derivatives: one for <span class="texhtml"><i>y</i> ≥ 0</span> and another for <span class="texhtml"><i>y</i> < 0</span>.

</p><p>It is substantially easier to implicitly differentiate the original equation:

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<annotation encoding="application/x-tex">{\displaystyle 4x^{3}+4y{\frac {dy}{dx}}=0\,,}</annotation>

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<p>giving

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}={\frac {-4x^{3}}{4y}}=-{\frac {x^{3}}{y}}\,.}</annotation>

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</math></span><img src="./44a422434c67cbb2d3596ff5f23ca3437d047d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.831ex; height:6.176ex;" alt="{\displaystyle {\frac {dy}{dx}}={\frac {-4x^{3}}{4y}}=-{\frac {x^{3}}{y}}\,.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Example_3">Example 3</h4></div>

<p>Often, it is difficult or impossible to solve explicitly for <span class="texhtml mvar" style="font-style:italic;">y</span>, and implicit differentiation is the only feasible method of differentiation. An example is the equation

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y^{5}-y=x\,.}</annotation>

</semantics>

</math></span><img src="./cdba7448951c537b7ca419f5595d044de57ce50e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.673ex; height:3.009ex;" alt="{\displaystyle y^{5}-y=x\,.}" loading="lazy"></span></dd></dl>

<p>It is impossible to <a href="Algebraic_expression" title="Algebraic expression">algebraically express</a> <span class="texhtml mvar" style="font-style:italic;">y</span> explicitly as a function of <span class="texhtml mvar" style="font-style:italic;">x</span>, and therefore one cannot find <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dy</i></span><span class="sr-only">/</span><span class="den"><i>dx</i></span></span>⁠</span></span> by explicit differentiation. Using the implicit method, <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dy</i></span><span class="sr-only">/</span><span class="den"><i>dx</i></span></span>⁠</span></span> can be obtained by differentiating the equation to obtain

</p>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5y^{4}{\frac {dy}{dx}}-{\frac {dy}{dx}}={\frac {dx}{dx}}\,,}</annotation>

</semantics>

</math></span><img src="./63a0223bd43376eb26f95ad961b176a65730b12f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.495ex; height:5.509ex;" alt="{\displaystyle 5y^{4}{\frac {dy}{dx}}-{\frac {dy}{dx}}={\frac {dx}{dx}}\,,}" loading="lazy"></span></dd></dl>

<p>where <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dx</i></span><span class="sr-only">/</span><span class="den"><i>dx</i></span></span>⁠</span> = 1</span>. Factoring out <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dy</i></span><span class="sr-only">/</span><span class="den"><i>dx</i></span></span>⁠</span></span> shows that

</p>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(5y^{4}-1\right){\frac {dy}{dx}}=1\,,}</annotation>

</semantics>

</math></span><img src="./886e1b3d27103278260407c144f170127340c82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.573ex; height:5.509ex;" alt="{\displaystyle \left(5y^{4}-1\right){\frac {dy}{dx}}=1\,,}" loading="lazy"></span></dd></dl>

<p>which yields the result

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}={\frac {1}{5y^{4}-1}}\,,}</annotation>

</semantics>

</math></span><img src="./74885de37e10de1e9e37f7bdcf2c8dd5225e89b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.73ex; height:6.009ex;" alt="{\displaystyle {\frac {dy}{dx}}={\frac {1}{5y^{4}-1}}\,,}" loading="lazy"></span></dd></dl>

<p>which is defined for

</p>

<mfrac>

<mroot>

</mrow>

</mroot>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mroot>

</mrow>

</mroot>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\neq \pm {\frac {1}{\sqrt[{4}]{5}}}\quad {\text{and}}\quad y\neq \pm {\frac {i}{\sqrt[{4}]{5}}}\,.}</annotation>

</semantics>

</math></span><img src="./02378d424997f05eb51116d1dd6119f4324b729b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.42ex; height:6.176ex;" alt="{\displaystyle y\neq \pm {\frac {1}{\sqrt[{4}]{5}}}\quad {\text{and}}\quad y\neq \pm {\frac {i}{\sqrt[{4}]{5}}}\,.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="General_formula_for_derivative_of_implicit_function">General formula for derivative of implicit function</h3></div>

<p>If <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>) = 0</span>, the derivative of the implicit function <span class="texhtml"><i>y</i>(<i>x</i>)</span> is given by<sup id="cite_ref-Stewart1998_2-0" class="reference"><a href="#cite_note-Stewart1998-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: §11.5">: §11.5 </span></sup>

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}=-{\frac {\,{\frac {\partial R}{\partial x}}\,}{\frac {\partial R}{\partial y}}}=-{\frac {R_{x}}{R_{y}}}\,,}</annotation>

</semantics>

</math></span><img src="./4465cbaeb2d99cc4bbb5be25d93b5d6f376ff92e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.627ex; height:8.843ex;" alt="{\displaystyle {\frac {dy}{dx}}=-{\frac {\,{\frac {\partial R}{\partial x}}\,}{\frac {\partial R}{\partial y}}}=-{\frac {R_{x}}{R_{y}}}\,,}" loading="lazy"></span></dd></dl>

<p>where <span class="texhtml"><i>R<sub>x</sub></i></span> and <span class="texhtml"><i>R<sub>y</sub></i></span> indicate the <a href="Partial_derivative" title="Partial derivative">partial derivatives</a> of <span class="texhtml mvar" style="font-style:italic;">R</span> with respect to <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>.

</p><p>The above formula comes from using the <a href="Chain_rule#Multivariable_case" title="Chain rule">generalized chain rule</a> to obtain the <a href="Total_derivative" title="Total derivative">total derivative</a> — with respect to <span class="texhtml mvar" style="font-style:italic;">x</span> — of both sides of <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>) = 0</span>:

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial R}{\partial x}}{\frac {dx}{dx}}+{\frac {\partial R}{\partial y}}{\frac {dy}{dx}}=0\,,}</annotation>

</semantics>

</math></span><img src="./13181ea39c36b09317409f9b6076edeb26f32f44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.735ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial R}{\partial x}}{\frac {dx}{dx}}+{\frac {\partial R}{\partial y}}{\frac {dy}{dx}}=0\,,}" loading="lazy"></span></dd></dl>

<p>hence

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial R}{\partial x}}+{\frac {\partial R}{\partial y}}{\frac {dy}{dx}}=0\,,}</annotation>

</semantics>

</math></span><img src="./c4af14acc33595334a6b5e9bb3fe3eac9dc88f03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.353ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial R}{\partial x}}+{\frac {\partial R}{\partial y}}{\frac {dy}{dx}}=0\,,}" loading="lazy"></span></dd></dl>

<p>which, when solved for <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dy</i></span><span class="sr-only">/</span><span class="den"><i>dx</i></span></span>⁠</span></span>, gives the expression above.

</p>

<div class="mw-heading mw-heading2"><h2 id="Implicit_function_theorem">Implicit function theorem</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Implicit_function_theorem" title="Implicit function theorem">Implicit function theorem</a></div>

<p>Let <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>)</span> be a <a href="Differentiable_function" title="Differentiable function">differentiable function</a> of two variables, and <span class="texhtml">(<i>a</i>, <i>b</i>)</span> be a pair of <a href="Real_number" title="Real number">real numbers</a> such that <span class="texhtml"><i>R</i>(<i>a</i>, <i>b</i>) = 0</span>. If <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num">∂<i>R</i></span><span class="sr-only">/</span><span class="den">∂<i>y</i></span></span>⁠</span> ≠ 0</span>, then <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>) = 0</span> defines an implicit function that is differentiable in some small enough <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of <span class="texhtml">(<i>a</i>, <i>b</i>)</span>; in other words, there is a differentiable function <span class="texhtml mvar" style="font-style:italic;">f</span> that is defined and differentiable in some neighbourhood of <span class="texhtml mvar" style="font-style:italic;">a</span>, such that <span class="texhtml"><i>R</i>(<i>x</i>, <i>f</i>(<i>x</i>)) = 0</span> for <span class="texhtml mvar" style="font-style:italic;">x</span> in this neighbourhood.

</p><p>The condition <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num">∂<i>R</i></span><span class="sr-only">/</span><span class="den">∂<i>y</i></span></span>⁠</span> ≠ 0</span> means that <span class="texhtml">(<i>a</i>, <i>b</i>)</span> is a <a href="Singular_point_of_a_curve" title="Singular point of a curve">regular point</a> of the <a href="Implicit_curve" title="Implicit curve">implicit curve</a> of implicit equation <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>) = 0</span> where the <a href="Tangent" title="Tangent">tangent</a> is not vertical.

</p><p>In a less technical language, implicit functions exist and can be differentiated, if the curve has a non-vertical tangent.<sup id="cite_ref-Stewart1998_2-1" class="reference"><a href="#cite_note-Stewart1998-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: §11.5">: §11.5 </span></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="In_algebraic_geometry">In algebraic geometry</h2></div>

<p>Consider a <a href="Relation_(mathematics)" title="Relation (mathematics)">relation</a> of the form <span class="texhtml"><i>R</i>(<i>x</i><sub>1</sub>, …, <i>x</i><sub><i>n</i></sub>) = 0</span>, where <span class="texhtml mvar" style="font-style:italic;">R</span> is a multivariable polynomial. The set of the values of the variables that satisfy this relation is called an <a href="Implicit_curve" title="Implicit curve">implicit curve</a> if <span class="texhtml"><i>n</i> = 2</span> and an <b><a href="Implicit_surface" title="Implicit surface">implicit surface</a></b> if <span class="texhtml"><i>n</i> = 3</span>. The implicit equations are the basis of <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, whose basic subjects of study are the simultaneous solutions of several implicit equations whose left-hand sides are polynomials. These sets of simultaneous solutions are called <a href="Affine_algebraic_set" class="mw-redirect" title="Affine algebraic set">affine algebraic sets</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="In_differential_equations">In differential equations</h2></div>

<p>The solutions of differential equations generally appear expressed by an implicit function.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications_in_economics">Applications in economics</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Marginal_rate_of_substitution">Marginal rate of substitution</h3></div>

<p>In <a href="Economics" title="Economics">economics</a>, when the level set <span class="texhtml"><i>R</i>(<i>x</i>, <i>y</i>) = 0</span> is an <a href="Indifference_curve" title="Indifference curve">indifference curve</a> for the quantities <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> consumed of two goods, the absolute value of the implicit derivative <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dy</i></span><span class="sr-only">/</span><span class="den"><i>dx</i></span></span>⁠</span></span> is interpreted as the <a href="Marginal_rate_of_substitution" title="Marginal rate of substitution">marginal rate of substitution</a> of the two goods: how much more of <span class="texhtml mvar" style="font-style:italic;">y</span> one must receive in order to be indifferent to a loss of one unit of <span class="texhtml mvar" style="font-style:italic;">x</span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Marginal_rate_of_technical_substitution">Marginal rate of technical substitution</h3></div>

<p>Similarly, sometimes the level set <span class="texhtml"><i>R</i>(<i>L</i>, <i>K</i>)</span> is an <a href="Isoquant" title="Isoquant">isoquant</a> showing various combinations of utilized quantities <span class="texhtml mvar" style="font-style:italic;">L</span> of labor and <span class="texhtml mvar" style="font-style:italic;">K</span> of <a href="Physical_capital" title="Physical capital">physical capital</a> each of which would result in the production of the same given quantity of output of some good. In this case the absolute value of the implicit derivative <span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"><i>dK</i></span><span class="sr-only">/</span><span class="den"><i>dL</i></span></span>⁠</span></span> is interpreted as the <a href="Marginal_rate_of_technical_substitution" title="Marginal rate of technical substitution">marginal rate of technical substitution</a> between the two factors of production: how much more capital the firm must use to produce the same amount of output with one less unit of labor.

</p>

<div class="mw-heading mw-heading3"><h3 id="Optimization">Optimization</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Mathematical_economics#Mathematical_optimization" title="Mathematical economics">Mathematical economics § Mathematical optimization</a></div>

<p>Often in <a href="Economic_theory" class="mw-redirect" title="Economic theory">economic theory</a>, some function such as a <a href="Utility_function" class="mw-redirect" title="Utility function">utility function</a> or a <a href="Profit_(economics)" title="Profit (economics)">profit</a> function is to be maximized with respect to a choice vector <span class="texhtml mvar" style="font-style:italic;">x</span> even though the objective function has not been restricted to any specific functional form. The <a href="Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a> guarantees that the <a href="First-order_condition" class="mw-redirect" title="First-order condition">first-order conditions</a> of the optimization define an implicit function for each element of the optimal vector <span class="texhtml"><i>x</i>*</span> of the choice vector <span class="texhtml mvar" style="font-style:italic;">x</span>. When profit is being maximized, typically the resulting implicit functions are the <a href="Labor_demand" title="Labor demand">labor demand</a> function and the <a href="Supply_function" class="mw-redirect" title="Supply function">supply functions</a> of various goods. When utility is being maximized, typically the resulting implicit functions are the <a href="Labor_supply" class="mw-redirect" title="Labor supply">labor supply</a> function and the <a href="Demand_function" class="mw-redirect" title="Demand function">demand functions</a> for various goods.

</p><p>Moreover, the influence of the problem's <a href="Parameter#Mathematical_functions" title="Parameter">parameters</a> on <span class="texhtml"><i>x</i>*</span> — the partial derivatives of the implicit function — can be expressed as <a href="Total_derivative" title="Total derivative">total derivatives</a> of the system of first-order conditions found using <a href="Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total differentiation</a>.

</p>

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<ul><li><a href="Implicit_curve" title="Implicit curve">Implicit curve</a></li>

<li><a href="Functional_equation" title="Functional equation">Functional equation</a></li>

<li><a href="Level_set" title="Level set">Level set</a>

<ul><li><a href="Contour_line" title="Contour line">Contour line</a></li>

<li><a href="Isosurface" title="Isosurface">Isosurface</a></li></ul></li>

<li><a href="Marginal_rate_of_substitution" title="Marginal rate of substitution">Marginal rate of substitution</a></li>

<li><a href="Implicit_function_theorem" title="Implicit function theorem">Implicit function theorem</a></li>

<li><a href="Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li>

<li><a href="Polygonizer" title="Polygonizer">Polygonizer</a></li>

<li><a href="Related_rates" title="Related rates">Related rates</a></li>

<li><a href="Folium_of_Descartes" title="Folium of Descartes">Folium of Descartes</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFChiang1984" class="citation book cs1"><a href="Alpha_Chiang" title="Alpha Chiang">Chiang, Alpha C.</a> (1984). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalmetho0000chia_b4p1"><i>Fundamental Methods of Mathematical Economics</i></a></span> (Third ed.). New York: McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-010813-7</bdi>.</cite></span>

</li>

<li id="cite_note-Stewart1998-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Stewart1998_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Stewart1998_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFStewart1998" class="citation book cs1">Stewart, James (1998). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/calculusconcepts00stew"><i>Calculus Concepts And Contexts</i></a></span>. Brooks/Cole Publishing Company. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-34330-9</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFKaplan2003" class="citation book cs1">Kaplan, Wilfred (2003). <i>Advanced Calculus</i>. Boston: Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-79937-5</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFBinmore1983" class="citation book cs1"><a href="Kenneth_Binmore" title="Kenneth Binmore">Binmore, K. G.</a> (1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K8RfQgAACAAJ&pg=PA198">"Implicit Functions"</a>. <i>Calculus</i>. New York: Cambridge University Press. pp. <span class="nowrap">198–</span>211. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-28952-1</bdi>.</cite></li>

<li><cite id="CITEREFRudin1976" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1976). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi"><i>Principles of Mathematical Analysis</i></a></span>. Boston: <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi/page/223">223–228</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-054235-X</bdi>.</cite></li>

<li><cite id="CITEREFSimonBlume1994" class="citation book cs1">Simon, Carl P.; <a href="Lawrence_E._Blume" title="Lawrence E. Blume">Blume, Lawrence</a> (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=l2nWMwEACAAJ&pg=PA334">"Implicit Functions and Their Derivatives"</a>. <i>Mathematics for Economists</i>. New York: W. W. Norton. pp. <span class="nowrap">334–</span>371. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-393-95733-0</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>Archived at <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211212/qb40J4N1fa4">Ghostarchive</a> and the <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170507005435/https://www.youtube.com/watch?v=qb40J4N1fa4">Wayback Machine</a>: <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=qb40J4N1fa4&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr">"Implicit Differentiation, What's Going on Here?"</a>. <i>3Blue1Brown</i>. Essence of Calculus. May 3, 2017 – via <a href="YouTube" title="YouTube">YouTube</a>.</cite></li></ul>

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<ul><li><a href="History_of_the_function_concept" title="History of the function concept">History</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types by domain and codomain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Constant_function" title="Constant function">Constant</a></li>

<li><a href="Identity_function" title="Identity function">Identity</a></li>

<li><a href="Linear_map" title="Linear map">Linear</a></li>

<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>

<li><a href="Rational_function" title="Rational function">Rational</a></li>

<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>

<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>

<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>

<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>

<li><a href="Injective_function" title="Injective function">Injective</a></li>

<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>

<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Function_composition" title="Function composition">Composition</a></li>

<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>

<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>

<li><a href="Partial_function" title="Partial function">Partial</a></li>

<li><a href="Function_space" title="Function space">Space</a></li>

<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>

<li><a href="Morphism" title="Morphism">Morphism</a></li>

<li><a href="Functor" title="Functor">Functor</a></li></ul>

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