commit 58aeb8993e627cbcfb494e7a1bd7b9d432010aed parent f352530246452cc5cc88a050b1571e364b6c384f Author: Ed van Bruggen <ed@edryd.org> Date: Wed, 4 Oct 2017 20:58:36 -0700 Add quotient rule post 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 @@ -0,0 +1,43 @@ +--- +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 + +$$\begin{align} + y & = \frac{u}{v} \\ + & = uv^{-1} \\ +y' & = v^{-1}u' + u(-v^{-2}v') \\ + & = \frac{u'}{v} - \frac{uv'}{v^2} \\ + & = \frac{v}{v}\cdot\frac{u'}{v} - \frac{uv'}{v^2} \\ + & = \frac{vu'}{v^2} - \frac{uv'}{v^2} \\ + & = \frac{vu' - uv'}{v^2} \\ +\end{align}$$