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commit 58aeb8993e627cbcfb494e7a1bd7b9d432010aed
parent f352530246452cc5cc88a050b1571e364b6c384f
Author: Ed van Bruggen <ed@edryd.org>
Date:   Wed,  4 Oct 2017 20:58:36 -0700

Diffstat:
_posts/2017-10-04-quotient-rule.md | 43+++++++++++++++++++++++++++++++++++++++++++

1 file changed, 43 insertions(+), 0 deletions(-)
diff --git a/_posts/2017-10-04-quotient-rule.md b/_posts/2017-10-04-quotient-rule.md
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+---
+title: "derivation of the quotient rule"
+tags: math calculus proof notes
+categories: math
+math: true
+---
+
+The quotient rule is used to take the derivative of a function with divided
+expressions:
+
+$$+\left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2} +$$
+
+It is possible to prove this rule by utilizing the definition of the
+derivative; however, this is not nearly as elegant as the following simple
+proofs which use other derivative properties instead.
+
+### natural logarithm
+
+\begin{align} + y & = \frac{u}{v} \\ +\mathrm{ln} y & = \mathrm{ln} \frac{u}{v} \\ + & = \mathrm{ln} u - \mathrm{ln} v \\ + \frac{y'}{y} & = \frac{u'}{u} - \frac{v'}{v} \\ + & = \frac{v}{v}\frac{u'}{u} - \frac{u}{u}\frac{v'}{v} \\ + & = \frac{vu' - uv'}{uv} \\ + y' & = y\frac{vu' - uv'}{uv} \\ + & = \frac{u}{v}\frac{vu' - uv'}{uv} \\ + & = \frac{vu' - uv'}{v^2} \\ +\end{align}
+
+### product rule
+
+\begin{align} + y & = \frac{u}{v} \\ + & = uv^{-1} \\ +y' & = v^{-1}u' + u(-v^{-2}v') \\ + & = \frac{u'}{v} - \frac{uv'}{v^2} \\ + & = \frac{v}{v}\cdot\frac{u'}{v} - \frac{uv'}{v^2} \\ + & = \frac{vu'}{v^2} - \frac{uv'}{v^2} \\ + & = \frac{vu' - uv'}{v^2} \\ +\end{align}