commit 39f7a643ffdc3a015914ff5fb80212ff38d0110b parent 1eff0ab13f66dc82f4acb338f1b4755c87f63fa1 Author: Ed van Bruggen <ed@edryd.org> Date: Thu, 21 Sep 2017 17:23:14 -0700 Publish logarithmic derivatives post Diffstat:
| _posts/2017-09-21-log-deriv.md | | | 44 | ++++++++++++++++++++++++++++++++++++++++++++ |
1 file changed, 44 insertions(+), 0 deletions(-)
diff --git a/_posts/2017-09-21-log-deriv.md b/_posts/2017-09-21-log-deriv.md @@ -0,0 +1,44 @@ +--- +title: "logarithmic proofs of derivative properties" +tags: math calculus proof notes +categories: math +math: true +--- + +### power rule + +$$\begin{align} + y & = x^n \\ +\mathrm{ln} y & = \mathrm{ln} x^n \\ + & = n \mathrm{ln} x \\ + \frac{y'}{y} & = n \frac{1}{x} \\ + y' & = yn \frac{1}{x} \\ + & = n x^n x^{-1} \\ + & = nx^{n-1}\\ +\end{align}$$ + +### product rule + +$$\begin{align} + y & = uv \\ +\mathrm{ln} y & = \mathrm{ln} uv \\ + & = \mathrm{ln} u + \mathrm{ln} v \\ + \frac{y'}{y} & = \frac{u'}{u} + \frac{v'}{v} \\ + y' & = y \frac{u'}{u} + \frac{v'}{v} \\ + & = uv \left(\frac{u'}{u} + \frac{v'}{v}\right) \\ + & = vu' + uv' \\ +\end{align}$$ + +### quotient rule + +$$\begin{align} + y & = \frac{u}{v} \\ +\mathrm{ln} y & = \mathrm{ln} \frac{u}{v} \\ + & = \mathrm{ln} u - \mathrm{ln} v \\ + \frac{y'}{y} & = \frac{u'}{u} - \frac{v'}{v} \\ + & = \frac{v}{v}\frac{u'}{u} - \frac{u}{u}\frac{v'}{v} \\ + & = \frac{vu' - uv'}{uv} \\ + y' & = y\frac{vu' - uv'}{uv} \\ + & = \frac{u}{v}\frac{vu' - uv'}{uv} \\ + & = \frac{vu' - uv'}{v^2} \\ +\end{align}$$