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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Differentiable function</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>differentiable function</b> of one <a href="Real_number" title="Real number">real</a> variable is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> whose <a href="Derivative" title="Derivative">derivative</a> exists at each point in its <a href="Domain_of_a_function" title="Domain of a function">domain</a>. In other words, the <a href="Graph_of_a_function" title="Graph of a function">graph</a> of a differentiable function has a non-<a href="Vertical_tangent" title="Vertical tangent">vertical</a> <a href="Tangent_line" class="mw-redirect" title="Tangent line">tangent line</a> at each interior point in its domain. A differentiable function is <a href="Smoothness" title="Smoothness">smooth</a> (the function is locally well approximated as a <a href="Linear_function" title="Linear function">linear function</a> at each interior point) and does not contain any break, angle, or <a href="Cusp_(singularity)" title="Cusp (singularity)">cusp</a>.

</p><p>If <span class="texhtml"><i>x</i><sub>0</sub></span> is an interior point in the domain of a function <span class="texhtml mvar" style="font-style:italic;">f</span>, then <span class="texhtml mvar" style="font-style:italic;">f</span> is said to be <i>differentiable at</i> <span class="texhtml"><i>x</i><sub>0</sub></span> if the derivative <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 f'(x_{0})}">

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</math></span><img src="./bc15f7bc4034ace9faccf92eb8e3f245541c5e6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.198ex; height:3.009ex;" alt="{\displaystyle f'(x_{0})}" loading="lazy"></span> exists. In other words, the graph of <span class="texhtml mvar" style="font-style:italic;">f</span> has a non-vertical tangent line at the point <span class="texhtml">(<i>x</i><sub>0</sub>, <i>f</i>(<i>x</i><sub>0</sub>))</span>. <span class="texhtml mvar" style="font-style:italic;">f</span> is said to be differentiable on <span class="texhtml mvar" style="font-style:italic;">U</span> if it is differentiable at every point of <span class="texhtml mvar" style="font-style:italic;">U</span>. <span class="texhtml mvar" style="font-style:italic;">f</span> is said to be <i>continuously differentiable</i> if its derivative is also a continuous function over the domain of the function <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="{\textstyle f}">

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<annotation encoding="application/x-tex">{\textstyle f}</annotation>

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</math></span><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy"></span>. Generally speaking, <span class="texhtml mvar" style="font-style:italic;">f</span> is said to be of class <em><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 C^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation>

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</math></span><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"></span></em> if its first <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 k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> derivatives <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="{\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}">

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<annotation encoding="application/x-tex">{\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}</annotation>

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</math></span><img src="./bb2aeaef7d0e6ea23fcea6d76b77f15fdd6f1361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.781ex; height:3.176ex;" alt="{\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}" loading="lazy"></span> exist and are continuous over the domain of the function <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="{\textstyle f}">

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<annotation encoding="application/x-tex">{\textstyle f}</annotation>

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</p><p>For a multivariable function, as shown <a href="#Differentiability_in_higher_dimensions">here</a>, the differentiability of it is something more complex than the existence of the partial derivatives of it.

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<div class="mw-heading mw-heading2"><h2 id="Differentiability_of_real_functions_of_one_variable">Differentiability of real functions of one variable</h2></div>

<p>A function <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 f:U\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle f:U\to \mathbb {R} }</annotation>

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</math></span><img src="./05f481901ff501baa824d1eab35eba6d9410ba57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.29ex; height:2.509ex;" alt="{\displaystyle f:U\to \mathbb {R} }" loading="lazy"></span>, defined on an open set <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="{\textstyle U\subset \mathbb {R} }">

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<annotation encoding="application/x-tex">{\textstyle U\subset \mathbb {R} }</annotation>

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</math></span><img src="./109e612a40380407c4dacd0cf68437e4b0e95485.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\textstyle U\subset \mathbb {R} }" loading="lazy"></span>, is said to be <i>differentiable</i> at <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 a\in U}">

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<annotation encoding="application/x-tex">{\displaystyle a\in U}</annotation>

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</math></span><img src="./cd1991ea9cb2ab076462a5538242321f0d0ee991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.853ex; height:2.176ex;" alt="{\displaystyle a\in U}" loading="lazy"></span> if the derivative

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<annotation encoding="application/x-tex">{\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}=\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}}</annotation>

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</math></span><img src="./b10ab00571bd3c31083b32e75c75e98f3432e53f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.783ex; height:5.843ex;" alt="{\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}=\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}}" loading="lazy"></span></dd></dl>

<p>exists. This implies that the function is <a href="Continuous_function" title="Continuous function">continuous</a> at <span class="texhtml mvar" style="font-style:italic;">a</span>.

</p><p>This function <span class="texhtml mvar" style="font-style:italic;">f</span> is said to be <i>differentiable</i> on <span class="texhtml mvar" style="font-style:italic;">U</span> if it is differentiable at every point of <span class="texhtml mvar" style="font-style:italic;">U</span>. In this case, the derivative of <span class="texhtml mvar" style="font-style:italic;">f</span> is thus a function from <span class="texhtml mvar" style="font-style:italic;">U</span> into <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 \mathbb {R} .}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation>

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</math></span><img src="./dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}" loading="lazy"></span>

</p><p>A continuous function is not necessarily differentiable, but a differentiable function is necessarily <a href="Continuous_function" title="Continuous function">continuous</a> (at every point where it is differentiable) as is shown below (in the section <a class="mw-selflink-fragment" href="#Differentiability_and_continuity">Differentiability and continuity</a>). A function is said to be <i>continuously differentiable</i> if its derivative is also a continuous function; there exist functions that are differentiable but not continuously differentiable (an example is given in the section <a class="mw-selflink-fragment" href="#Differentiability_classes">Differentiability classes</a>).

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

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

<p>The above definition can be extended to define the derivative at <a href="Boundary_(topology)" title="Boundary (topology)">boundary points</a>. The derivative of a function <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="{\textstyle f:A\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\textstyle f:A\to \mathbb {R} }</annotation>

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</math></span><img src="./b0343a2a773b8a10efb86b14596955d9ad447f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.251ex; height:2.509ex;" alt="{\textstyle f:A\to \mathbb {R} }" loading="lazy"></span> defined on a closed subset <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="{\textstyle A\subsetneq \mathbb {R} }">

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<annotation encoding="application/x-tex">{\textstyle A\subsetneq \mathbb {R} }</annotation>

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</math></span><img src="./ff95a77326d8b245a988f9e80f7daa78b611c9c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.52ex; height:2.676ex;" alt="{\textstyle A\subsetneq \mathbb {R} }" loading="lazy"></span> of the real numbers, evaluated at a boundary point <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="{\textstyle c}">

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<annotation encoding="application/x-tex">{\textstyle c}</annotation>

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</math></span><img src="./7d411ca19645ddd4fff0704de95ec770681093bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\textstyle c}" loading="lazy"></span>, can be defined as the following one-sided limit, where the argument <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="{\textstyle x}">

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<annotation encoding="application/x-tex">{\textstyle x}</annotation>

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</math></span><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}" loading="lazy"></span> approaches <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="{\textstyle c}">

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<annotation encoding="application/x-tex">{\textstyle c}</annotation>

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</math></span><img src="./7d411ca19645ddd4fff0704de95ec770681093bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\textstyle c}" loading="lazy"></span> such that it is always within <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="{\textstyle A}">

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<annotation encoding="application/x-tex">{\textstyle A}</annotation>

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</math></span><img src="./a118c6ad00742b3f5dccd2f0e74b5e369df6fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle f'(c)=\lim _{\scriptstyle x\to c \atop \scriptstyle x\in A}{\frac {f(x)-f(c)}{x-c}}.}</annotation>

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</math></span><img src="./5bcacc5ba56b7448deedabec4c9311c11cb73c16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.995ex; height:7.176ex;" alt="{\displaystyle f'(c)=\lim _{\scriptstyle x\to c \atop \scriptstyle x\in A}{\frac {f(x)-f(c)}{x-c}}.}" loading="lazy"></span></dd></dl>

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</math></span><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}" loading="lazy"></span> to remain within <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="{\textstyle A}">

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<annotation encoding="application/x-tex">{\textstyle A}</annotation>

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</math></span><img src="./a118c6ad00742b3f5dccd2f0e74b5e369df6fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}" loading="lazy"></span>, which is a subset of the reals, it follows that this limit will be defined as either

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<annotation encoding="application/x-tex">{\displaystyle f'(c)=\lim _{x\to c^{+}}{\frac {f(x)-f(c)}{x-c}}\quad {\text{or}}\quad f'(c)=\lim _{x\to c^{-}}{\frac {f(x)-f(c)}{x-c}}.}</annotation>

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</math></span><img src="./c7da69ba83a9efc56f95a26bf94156ac5c5f7af5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:57.353ex; height:6.176ex;" alt="{\displaystyle f'(c)=\lim _{x\to c^{+}}{\frac {f(x)-f(c)}{x-c}}\quad {\text{or}}\quad f'(c)=\lim _{x\to c^{-}}{\frac {f(x)-f(c)}{x-c}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Differentiability_and_continuity">Differentiability and continuity</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Continuous_function" title="Continuous function">Continuous function</a></div>

<p>If <span class="texhtml"><i>f</i></span> is differentiable at a point <span class="texhtml"><i>x</i><sub>0</sub></span>, then <span class="texhtml"><i>f</i></span> must also be <a href="Continuous_function" title="Continuous function">continuous</a> at <span class="texhtml"><i>x</i><sub>0</sub></span>. In particular, any differentiable function must be continuous at every point in its domain. <i>The converse does not hold</i>: a continuous function need not be differentiable. For example, a function with a bend, <a href="Cusp_(singularity)" title="Cusp (singularity)">cusp</a>, or <a href="Vertical_tangent" title="Vertical tangent">vertical tangent</a> may be continuous, but fails to be differentiable at the location of the anomaly.

</p><p>Most functions that occur in practice have derivatives at all points or at <a href="Almost_everywhere" title="Almost everywhere">almost every</a> point. However, a result of <a href="Stefan_Banach" title="Stefan Banach">Stefan Banach</a> states that the set of functions that have a derivative at some point is a <a href="Meagre_set" title="Meagre set">meagre set</a> in the space of all continuous functions.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the <a href="Weierstrass_function" title="Weierstrass function">Weierstrass function</a>.

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<div class="mw-heading mw-heading2"><h2 id="Differentiability_classes">Differentiability classes</h2></div>

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

<p>A function <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="{\textstyle f}">

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</math></span><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy"></span> is said to be <em><style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style><span class="vanchor"><span class="vanchor-text">continuously differentiable</span></span></em> if the derivative <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="{\textstyle f^{\prime }(x)}">

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<annotation encoding="application/x-tex">{\textstyle f^{\prime }(x)}</annotation>

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</math></span><img src="./b9ff3eaec32535095ec3843df707486aca3402c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:2.843ex;" alt="{\textstyle f^{\prime }(x)}" loading="lazy"></span> exists and is itself a continuous function. Although the derivative of a differentiable function never has a <a href="Jump_discontinuity" class="mw-redirect" title="Jump discontinuity">jump discontinuity</a>, it is possible for the derivative to have an <a href="Classification_of_discontinuities#Essential_discontinuity" title="Classification of discontinuities">essential discontinuity</a>. For example, the function

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<annotation encoding="application/x-tex">{\displaystyle f(x)\;=\;{\begin{cases}x^{2}\sin(1/x)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}}</annotation>

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</math></span></span>

is differentiable at 0, since

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<annotation encoding="application/x-tex">{\displaystyle f'(0)=\lim _{\varepsilon \to 0}\left({\frac {\varepsilon ^{2}\sin(1/\varepsilon )-0}{\varepsilon }}\right)=0}</annotation>

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</math></span></span>

exists. However, for <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\neq 0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq 0,}</annotation>

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</math></span><img src="./6738efbc0e8285d9fc72fa299b75f7fa903ee95d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.237ex; height:2.676ex;" alt="{\displaystyle x\neq 0,}" loading="lazy"></span> <a href="Differentiation_rules" title="Differentiation rules">differentiation rules</a> imply

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<annotation encoding="application/x-tex">{\displaystyle f'(x)=2x\sin(1/x)-\cos(1/x)\;,}</annotation>

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which has no limit as <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\to 0.}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x\to 0.}</annotation>

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</math></span><img src="./68a3bd180bdd34b85ad7df6d774809bcc47fdb71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.753ex; height:2.176ex;" alt="{\displaystyle x\to 0.}" loading="lazy"></span> Thus, this example shows the existence of a function that is differentiable but not continuously differentiable (i.e., the derivative is not a continuous function). Nevertheless, <a href="Darboux's_theorem_(analysis)" title="Darboux's theorem (analysis)">Darboux's theorem</a> implies that the derivative of any function satisfies the conclusion of the <a href="Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a>.

</p><p>Similarly to how <a href="Continuous_function" title="Continuous function">continuous functions</a> are said to be of <em>class <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 C^{0},}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle C^{0},}</annotation>

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</math></span><img src="./98d54cd3123110b170d179fda61f57b1c24da5ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.499ex; height:3.009ex;" alt="{\displaystyle C^{0},}" loading="lazy"></span></em> continuously differentiable functions are sometimes said to be of <em>class <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 C^{1}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation>

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</math></span><img src="./bd24bae0d7570018e828e19851902c09c618af91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" loading="lazy"></span></em>. A function is of <em>class <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 C^{2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle C^{2}}</annotation>

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</math></span><img src="./4fd6a5946b7e916352b0afc557f992328bac85e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{2}}" loading="lazy"></span></em> if the first and <a href="Second_derivative" title="Second derivative">second derivative</a> of the function both exist and are continuous. More generally, a function is said to be of <em>class <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 C^{k}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation>

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</math></span><img src="./167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"></span></em> if the first <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 k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> derivatives <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="{\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}">

<msup>

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<annotation encoding="application/x-tex">{\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}</annotation>

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</math></span><img src="./bb2aeaef7d0e6ea23fcea6d76b77f15fdd6f1361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.781ex; height:3.176ex;" alt="{\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}" loading="lazy"></span> all exist and are continuous. If derivatives <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 f^{(n)}}">

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{(n)}}</annotation>

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</math></span><img src="./9dfb1963ccde0e87eb3838f51dc19041e2ff3816.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.818ex; height:3.176ex;" alt="{\displaystyle f^{(n)}}" loading="lazy"></span> exist for all positive integers <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="{\textstyle n,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n,}</annotation>

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</math></span><img src="./6da6c2c1a406c3c6c97a12a5b21a0e6d92f7f935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\textstyle n,}" loading="lazy"></span> the function is <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> or equivalently, of <em>class <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 C^{\infty }.}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }.}</annotation>

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</math></span><img src="./149f006dc6f05e8f662e839f68194d287119da68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.32ex; height:2.343ex;" alt="{\displaystyle C^{\infty }.}" loading="lazy"></span></em>

</p>

<div class="mw-heading mw-heading2"><h2 id="Differentiability_in_higher_dimensions">Differentiability in higher dimensions</h2></div>

<p>A <a href="Function_of_several_real_variables" title="Function of several real variables">function of several real variables</a> <span class="texhtml"><b>f</b>: <b>R</b><sup><i>m</i></sup> → <b>R</b><sup><i>n</i></sup></span> is said to be differentiable at a point <span class="texhtml"><b>x</b><sub>0</sub></span> if <a href="There_exists" class="mw-redirect" title="There exists">there exists</a> a <a href="Linear_map" title="Linear map">linear map</a> <span class="texhtml"><b>J</b>: <b>R</b><sup><i>m</i></sup> → <b>R</b><sup><i>n</i></sup></span> such that

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<annotation encoding="application/x-tex">{\displaystyle \lim _{\mathbf {h} \to \mathbf {0} }{\frac {\|\mathbf {f} (\mathbf {x_{0}} +\mathbf {h} )-\mathbf {f} (\mathbf {x_{0}} )-\mathbf {J} \mathbf {(h)} \|_{\mathbf {R} ^{n}}}{\|\mathbf {h} \|_{\mathbf {R} ^{m}}}}=0.}</annotation>

</semantics>

</math></span><img src="./268352e5f56cde57bf51a8965dda4e9c3203644c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.561ex; height:6.509ex;" alt="{\displaystyle \lim _{\mathbf {h} \to \mathbf {0} }{\frac {\|\mathbf {f} (\mathbf {x_{0}} +\mathbf {h} )-\mathbf {f} (\mathbf {x_{0}} )-\mathbf {J} \mathbf {(h)} \|_{\mathbf {R} ^{n}}}{\|\mathbf {h} \|_{\mathbf {R} ^{m}}}}=0.}" loading="lazy"></span></dd></dl>

<p>If a function is differentiable at <span class="texhtml"><b>x</b><sub>0</sub></span>, then all of the <a href="Partial_derivative" title="Partial derivative">partial derivatives</a> exist at <span class="texhtml"><b>x</b><sub>0</sub></span>, and the linear map <span class="texhtml"><b>J</b></span> is given by the <a href="Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a>, an <i>n</i> × <i>m</i> matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the <a href="Fundamental_increment_lemma" title="Fundamental increment lemma">fundamental increment lemma</a> found in single-variable calculus.

</p><p>If all the partial derivatives of a function exist in a <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighborhood</a> of a point <span class="texhtml"><b>x</b><sub>0</sub></span> and are continuous at the point <span class="texhtml"><b>x</b><sub>0</sub></span>, then the function is differentiable at that point <span class="texhtml"><b>x</b><sub>0</sub></span>.

</p><p>However, the existence of the partial derivatives (or even of all the <a href="Directional_derivative" title="Directional derivative">directional derivatives</a>) does not guarantee that a function is differentiable at a point. For example, the function <span class="texhtml"><i>f</i>: <b>R</b><sup>2</sup> → <b>R</b></span> defined by

</p>

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)={\begin{cases}x&{\text{if }}y\neq x^{2}\\0&{\text{if }}y=x^{2}\end{cases}}}</annotation>

</semantics>

</math></span></span></dd></dl>

<p>is not differentiable at <span class="texhtml">(0, 0)</span>, but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)={\begin{cases}y^{3}/(x^{2}+y^{2})&{\text{if }}(x,y)\neq (0,0)\\0&{\text{if }}(x,y)=(0,0)\end{cases}}}</annotation>

</semantics>

</math></span><img src="./ba3df8cc6421be6baf3602e6a87836917d2d4f77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.683ex; height:6.176ex;" alt="{\displaystyle f(x,y)={\begin{cases}y^{3}/(x^{2}+y^{2})&{\text{if }}(x,y)\neq (0,0)\\0&{\text{if }}(x,y)=(0,0)\end{cases}}}" loading="lazy"></span></dd></dl>

<p>is not differentiable at <span class="texhtml">(0, 0)</span>, but again all of the partial derivatives and directional derivatives exist.

</p>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a> and <a href="Smoothness#Multivariate_differentiability_classes" title="Smoothness">Smoothness § Multivariate differentiability classes</a></div>

<div class="mw-heading mw-heading2"><h2 id="Differentiability_in_complex_analysis">Differentiability in complex analysis</h2></div>

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

<p>In <a href="Complex_analysis" title="Complex analysis">complex analysis</a>, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing <a href="Complex_number" title="Complex number">complex numbers</a>. So, a function <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="{\textstyle f:\mathbb {C} \to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\textstyle f:\mathbb {C} \to \mathbb {C} }</annotation>

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</math></span><img src="./47f0927311aba8143fae775819d601fcf512f01a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\textstyle f:\mathbb {C} \to \mathbb {C} }" loading="lazy"></span> is said to be differentiable at <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="{\textstyle x=a}">

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<annotation encoding="application/x-tex">{\textstyle x=a}</annotation>

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</math></span><img src="./70e0fe729bde9223f1b509db0471471c21261b82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\textstyle x=a}" loading="lazy"></span> when

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<annotation encoding="application/x-tex">{\displaystyle f'(a)=\lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {f(a+h)-f(a)}{h}}.}</annotation>

</semantics>

</math></span><img src="./52fef41871e068e45afc1f52bbaf0397ffe9d992.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:29.08ex; height:7.343ex;" alt="{\displaystyle f'(a)=\lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {f(a+h)-f(a)}{h}}.}" loading="lazy"></span></dd></dl>

<p>Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function <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="{\textstyle f:\mathbb {C} \to \mathbb {C} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:\mathbb {C} \to \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./47f0927311aba8143fae775819d601fcf512f01a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\textstyle f:\mathbb {C} \to \mathbb {C} }" loading="lazy"></span>, that is complex-differentiable at a point <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="{\textstyle x=a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x=a}</annotation>

</semantics>

</math></span><img src="./70e0fe729bde9223f1b509db0471471c21261b82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\textstyle x=a}" loading="lazy"></span> is automatically differentiable at that point, when viewed as a function <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 f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}</annotation>

</semantics>

</math></span><img src="./25c44be73c93794e647a2a66e3fc18f527395064.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.294ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}" loading="lazy"></span>. This is because the complex-differentiability implies that

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</munder>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {|f(a+h)-f(a)-f'(a)h|}{|h|}}=0.}</annotation>

</semantics>

</math></span><img src="./91b847626163a9bce95419a382df0227856c448b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:35.715ex; height:7.343ex;" alt="{\displaystyle \lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {|f(a+h)-f(a)-f'(a)h|}{|h|}}=0.}" loading="lazy"></span></dd></dl>

<p>However, a function <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="{\textstyle f:\mathbb {C} \to \mathbb {C} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:\mathbb {C} \to \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./47f0927311aba8143fae775819d601fcf512f01a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\textstyle f:\mathbb {C} \to \mathbb {C} }" loading="lazy"></span> can be differentiable as a multi-variable function, while not being complex-differentiable. For example, <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 f(z)={\frac {z+{\overline {z}}}{2}}}">

<mfrac>

<mrow>

<mover>

</mover>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {z+{\overline {z}}}{2}}}</annotation>

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</math></span><img src="./0d2aa2cf99fd9c9e4052ed8b119064f688609784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.247ex; height:5.509ex;" alt="{\displaystyle f(z)={\frac {z+{\overline {z}}}{2}}}" loading="lazy"></span> is differentiable at every point, viewed as the 2-variable <a href="Real-valued_function" title="Real-valued function">real function</a> <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 f(x,y)=x}">

</mstyle>

</mrow>

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

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</math></span><img src="./c6bec423b26f491e42ada1e0cface4f4f396471d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.035ex; height:2.843ex;" alt="{\displaystyle f(x,y)=x}" loading="lazy"></span>, but it is not complex-differentiable at any point because the limit <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="{\textstyle \lim _{h\to 0}{\frac {h+{\bar {h}}}{2h}}}">

</mrow>

</munder>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

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</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\textstyle \lim _{h\to 0}{\frac {h+{\bar {h}}}{2h}}}</annotation>

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</p><p>Any function that is complex-differentiable in a neighborhood of a point is called <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> at that point. Such a function is necessarily infinitely differentiable, and in fact <a href="Analytic_function" title="Analytic function">analytic</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Differentiable_functions_on_manifolds">Differentiable functions on manifolds</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Differentiable_manifold#Differentiable_functions" title="Differentiable manifold">Differentiable manifold § Differentiable functions</a></div>

<p>If <i>M</i> is a <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>, a real or complex-valued function <i>f</i> on <i>M</i> is said to be differentiable at a point <i>p</i> if it is differentiable with respect to some (or any) coordinate chart defined around <i>p</i>. If <i>M</i> and <i>N</i> are differentiable manifolds, a function <i>f</i>: <i>M</i> → <i>N</i> is said to be differentiable at a point <i>p</i> if it is differentiable with respect to some (or any) coordinate charts defined around <i>p</i> and <i>f</i>(<i>p</i>).

</p>

<ul><li><a href="Generalizations_of_the_derivative" title="Generalizations of the derivative">Generalizations of the derivative</a></li>

<li><a href="Semi-differentiability" title="Semi-differentiability">Semi-differentiability</a></li>

<li><a href="Differentiable_programming" title="Differentiable programming">Differentiable programming</a></li></ul>

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

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</style><cite id="CITEREFBanach1931" class="citation journal cs1">Banach, S. (1931). <a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Fsm-3-1-174-179">"Über die Baire'sche Kategorie gewisser Funktionenmengen"</a>. <i><a href="Studia_Mathematica" title="Studia Mathematica">Studia Math.</a></i> <b>3</b> (1): <span class="nowrap">174–</span>179. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Fsm-3-1-174-179">10.4064/sm-3-1-174-179</a></span>.</cite>. Cited by <cite id="CITEREFHewitt,_EStromberg,_K1963" class="citation book cs1">Hewitt, E; Stromberg, K (1963). <i>Real and abstract analysis</i>. Springer-Verlag. Theorem 17.8.</cite></span>

</li>

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</style><div id="Differentiable_computing254" style="font-size:114%;margin:0 4em">Differentiable computing</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><b><a href="Differentiable_programming" title="Differentiable programming">Differentiable programming</a></b></li>

<li><a href="Information_geometry" title="Information geometry">Information geometry</a></li>

<li><a href="Statistical_manifold" title="Statistical manifold">Statistical manifold</a></li>

<li><a href="Automatic_differentiation" title="Automatic differentiation">Automatic differentiation</a></li>

<li><a href="Neuromorphic_computing" title="Neuromorphic computing">Neuromorphic computing</a></li>

<li><a href="Pattern_recognition" title="Pattern recognition">Pattern recognition</a></li>

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li>

<li><a href="Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li>

<li><a href="Inductive_bias" title="Inductive bias">Inductive bias</a></li></ul>

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

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

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software libraries</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="TensorFlow" title="TensorFlow">TensorFlow</a></li>

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

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

<li><a href="Scikit-learn" title="Scikit-learn">scikit-learn</a></li>

<li><a href="Theano_(software)" title="Theano (software)">Theano</a></li>

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

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"></span> Portals

<ul><li><a href="Portal%3AComputer_programming" title="Portal:Computer programming">Computer programming</a></li>

<li><a href="Portal%3ATechnology" title="Portal:Technology">Technology</a></li></ul></li></ul>

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