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Diffstat (limited to 'mindmap/derivative.org')
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diff --git a/mindmap/derivative.org b/mindmap/derivative.org index be84116..d046459 100644 --- a/mindmap/derivative.org +++ b/mindmap/derivative.org @@ -7,7 +7,6 @@ #+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> #+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> #+options: broken-links:t -#+OPTIONS: tex:dvipng * Derivation Let's say we want to know the rate of change of the [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] \(f(x) = x^{2}\). Because this [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] is not @@ -76,14 +75,17 @@ We derive many of them here. = \frac{d}{dx}f(x) + \frac{d}{dx}g(x) \end{align*} of course, subtraction works in the same way. -** Multiplication Rule +** product rule +:PROPERTIES: +:ID: d1e245f4-0b04-450e-8465-a9c85fe57f7e +:END: \begin{align*} \frac{d}{dx}(f(x)g(x)) = \lim_{h\to0}\frac{f(x + h)g(x + h) - f(x)g(x)}{h} = \lim_{h\to0}\frac{f(x + h)g(x + h) - f(x)g(x + h) + f(x)g(x + h) - f(x)g(x)}{h} \\ = \lim_{h\to0}\frac{g(x + h)(f(x + h) - f(x)) + f(x)(g(x + h) - g(x))}{h} \\ = g(x)\lim_{h\to0}\frac{f(x + h) - f(x)}{h} + f(x)\frac{g(x + h) - g(x)}{h} = g(x)f'(x) + g'(x)f(x) \end{align*} And using the this rule as well as the chain rule and power rule which we will show later, the division rule is easily acquired. -** Chain Rule +** chain rule :PROPERTIES: :ID: ffd1bc3d-ab64-4916-9c09-0c89d2731b6d :END: |