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class="breadcrumbs__item"><a class="breadcrumbs__link" itemprop="item" href="/category/math-refresher">Math Refresher</a><meta itemprop="position" content="1"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active">Basics<meta itemprop="position" content="2"></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><header><h1>Basics</h1></header><h2 class="anchor anchorWithStickyNavbar_LWe7" id="basic-derivatives">Basic Derivatives<a href="#basic-derivatives" class="hash-link" aria-label="Direct link to Basic Derivatives" title="Direct link to Basic Derivatives"></a></h2><table><thead><tr><th align="center">Function</th><th align="center">Derivative</th></tr></thead><tbody><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">f(x) = C</annotation></semantics></math>f(x)=C (constant)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(x) = 0</annotation></semantics></math>f′(x)=0</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">f(x) = x^n</annotation></semantics></math>f(x)=xn</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f'(x) = nx^{n-1}</annotation></semantics></math>f′(x)=nxn−1</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy="false">(</mo><mi>C</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = 1-exp(Cx)</annotation></semantics></math>f(x)=1−exp(Cx)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>C</mi><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>C</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x) = -C \exp(Cx)</annotation></semantics></math>f′(x)=−Cexp(Cx)</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = \exp(x)</annotation></semantics></math>f(x)=exp(x)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x) = \exp(x)</annotation></semantics></math>f′(x)=exp(x)</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = \exp(f(x))</annotation></semantics></math>f(x)=exp(f(x))</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x) = \exp(f(x)) \cdot f'(x)</annotation></semantics></math>f′(x)=exp(f(x))⋅f′(x)</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = \log(x)</annotation></semantics></math>f(x)=log(x)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><annotation encoding="application/x-tex">f'(x) = \frac{1}{x}</annotation></semantics></math>f′(x)=x1</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = \log(C)</annotation></semantics></math>f(x)=log(C)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(x) = 0</annotation></semantics></math>f′(x)=0</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = g(h(x))</annotation></semantics></math>f(x)=g(h(x))</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>g</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>h</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x) = g'(h(x)) \cdot h'(x)</annotation></semantics></math>f′(x)=g′(h(x))⋅h′(x)</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>v</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = u(x) \cdot v(x)</annotation></semantics></math>f(x)=u(x)⋅v(x)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>u</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>v</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>v</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)</annotation></semantics></math>f′(x)=u′(x)⋅v(x)+u(x)⋅v′(x)</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">∣</mo><mi>x</mi><mo stretchy="false">∣</mo></mrow><annotation encoding="application/x-tex">f(x) = \lvert x \rvert</annotation></semantics></math>f(x)=∣x∣</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mi>i</mi><mi>g</mi><mi>n</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x) = sign(1)</annotation></semantics></math>f′(x)=sign(1)</td></tr></tbody></table><h2 class="anchor anchorWithStickyNavbar_LWe7" id="common-ml-functions">Common ML Functions<a href="#common-ml-functions" class="hash-link" aria-label="Direct link to Common ML Functions" title="Direct link to Common ML Functions"></a></h2><ul><li>Logarithm and Exponential:<table><thead><tr><th align="center">Logarithm Rules</th><th align="center">Exponential Rules</th></tr></thead><tbody><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ln(ab) = \ln(a) + \ln(b)</annotation></semantics></math>ln(ab)=ln(a)+ln(b)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>a</mi></msup><mo>⋅</mo><msup><mi>e</mi><mi>b</mi></msup><mo>=</mo><msup><mi>e</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></msup></mrow><annotation encoding="application/x-tex">e^a \cdot e^b = e^{a+b}</annotation></semantics></math>ea⋅eb=ea+b</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mfrac><mi>a</mi><mi>b</mi></mfrac><mo fence="true">)</mo></mrow><mo>=</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)</annotation></semantics></math>ln(ba)=ln(a)−ln(b)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>e</mi><mi>a</mi></msup><msup><mi>e</mi><mi>b</mi></msup></mfrac><mo>=</mo><msup><mi>e</mi><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{e^a}{e^b} = e^{a-b}</annotation></semantics></math>ebea=ea−b</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mi>a</mi><mi>b</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ln(a^b) = b \ln(a)</annotation></semantics></math>ln(ab)=bln(a)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>e</mi><mi>a</mi></msup><msup><mo stretchy="false">)</mo><mi>b</mi></msup><mo>=</mo><msup><mi>e</mi><mrow><mi>a</mi><mi>b</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(e^a)^b = e^{ab}</annotation></semantics></math>(ea)b=eab</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\ln(e) = 1</annotation></semantics></math>ln(e)=1</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">e^0 = 1</annotation></semantics></math>e0=1</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\ln(1) = 0</annotation></semantics></math>ln(1)=0</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">e^{\ln(x)} = x</annotation></semantics></math>eln(x)=x</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\ln(e^x) = x</annotation></semantics></math>ln(ex)=x</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mi>a</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">e^{\ln(a) + \ln(b)} = ab</annotation></semantics></math>eln(a)+ln(b)=ab</td></tr><tr><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><msub><mo>∏</mo><mi>i</mi></msub><msub><mi>a</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mi>i</mi></msub><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ln\left(\prod_{i} a_i\right) = \sum_{i} \ln(a_i)</annotation></semantics></math>ln(∏iai)=∑iln(ai)</td><td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>∏</mo><mi>i</mi></msub><msup><mi>e</mi><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msup><mi>e</mi><mrow><msub><mo>∑</mo><mi>i</mi></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\prod_{i} e^{f(x_i)} = e^{\sum_{i} f(x_i)}</annotation></semantics></math>∏ief(xi)=e∑if(xi)</td></tr></tbody></table></li><li>Sigmoid Function:</li></ul><div class="math math-display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\sigma(x) = \frac{1}{1 + e^{-x}}</annotation></semantics></math>σ(x)=1+e−x1</div><div class="math math-display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>σ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma'(x) = \sigma(x)(1 - \sigma(x))</annotation></semantics></math>σ′(x)=σ(x)(1−σ(x))</div><ul><li>Softmax Function (for class <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>i):</li></ul><div class="math math-display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>softmax</mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><msup><mi>e</mi><msub><mi>x</mi><mi>i</mi></msub></msup><mrow><munder><mo>∑</mo><mi>j</mi></munder><msup><mi>e</mi><msub><mi>x</mi><mi>j</mi></msub></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text{softmax}(x_i) = \frac{e^{x_i}}{\sum_{j} e^{x_j}}</annotation></semantics></math>softmax(xi)=∑jexjexi</div><ul><li>Log-Likelihood:</li></ul><div class="math math-display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{d}{dx}\ln(f(x)) = \frac{f'(x)}{f(x)}</annotation></semantics></math>dxdln(f(x))=f(x)f′(x)</div><h2 class="anchor anchorWithStickyNavbar_LWe7" id="gradient">Gradient<a href="#gradient" class="hash-link" aria-label="Direct link to Gradient" title="Direct link to Gradient"></a></h2>The gradient is a vector of all partial derivatives:<div class="math math-display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\nabla f = \begin{bmatrix} \frac{\partial f}{\partial x_1} \\ \frac{\partial f}{\partial x_2} \\ \vdots \\ \frac{\partial f}{\partial x_n} \end{bmatrix}</annotation></semantics></math>∇f=⎣<svg xmlns="http://www.w3.org/2000/svg" width="0.6667em" height="2.416em" style="width:0.6667em" viewBox="0 0 666.67 2416" preserveAspectRatio="xMinYMin"><path d="M319 0 H403 V2416 H319z M319 0 H403 V2416 H319z"></path></svg>⎡∂x1∂f∂x2∂f⋮∂xn∂f⎦<svg xmlns="http://www.w3.org/2000/svg" width="0.6667em" height="2.416em" style="width:0.6667em" viewBox="0 0 666.67 2416" preserveAspectRatio="xMinYMin"><path d="M263 0 H347 V2416 H263z M263 0 H347 V2416 H263z"></path></svg>⎤</div>The gradient points in the direction of steepest ascent, which is why gradient descent moves in the negative gradient direction to minimize loss functions.</div></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Docs pages"><a class="pagination-nav__link pagination-nav__link--prev" href="/category/math-refresher"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Math Refresher</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/MachineLearning/BasicMath/LinearAlgebra"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Linear 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