commit e79112262aa4b6f90373d1e6da825bda30f04559
parent bb1d2d36571da252d1cccc0c41ba797e164c2b44
Author: Ed van Bruggen <edvb@uw.edu>
Date: Sun, 1 Jul 2018 00:36:43 -0700
Combine derivative posts
Diffstat:
3 files changed, 74 insertions(+), 87 deletions(-)
diff --git a/_posts/2017-09-21-log-deriv.md b/_posts/2017-09-21-log-deriv.md
@@ -1,44 +0,0 @@
----
-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
-
-$$\begin{align}
- y & = \frac{u}{v} \\
-\mathrm{ln} y & = \mathrm{ln} \frac{u}{v} \\
- & = \mathrm{ln} u - \mathrm{ln} v \\
- \frac{y'}{y} & = \frac{u'}{u} - \frac{v'}{v} \\
- & = \frac{v}{v}\frac{u'}{u} - \frac{u}{u}\frac{v'}{v} \\
- & = \frac{vu' - uv'}{uv} \\
- y' & = y\frac{vu' - uv'}{uv} \\
- & = \frac{u}{v}\frac{vu' - uv'}{uv} \\
- & = \frac{vu' - uv'}{v^2} \\
-\end{align}$$
diff --git a/_posts/2017-10-04-deriv-proofs.md b/_posts/2017-10-04-deriv-proofs.md
@@ -0,0 +1,74 @@
+---
+title: "proof of derivative properties"
+tags: math calculus proof notes
+categories: math
+math: true
+---
+
+## derivation of the quotient rule
+
+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.
+
+### product rule
+
+$$\begin{align}
+ y & = \frac{u}{v} \\
+ & = uv^{-1} \\
+y' & = v^{-1}u' + u(-v^{-2}v') \\
+ & = \frac{u'}{v} - \frac{uv'}{v^2} \\
+ & = \frac{v}{v}\cdot\frac{u'}{v} - \frac{uv'}{v^2} \\
+ & = \frac{vu'}{v^2} - \frac{uv'}{v^2} \\
+ & = \frac{vu' - uv'}{v^2} \\
+\end{align}$$
+
+### logarithm
+
+$$\begin{align}
+ y & = \frac{u}{v} \\
+\mathrm{ln} y & = \mathrm{ln} \frac{u}{v} \\
+ & = \mathrm{ln} u - \mathrm{ln} v \\
+ \frac{y'}{y} & = \frac{u'}{u} - \frac{v'}{v} \\
+ & = \frac{v}{v}\frac{u'}{u} - \frac{u}{u}\frac{v'}{v} \\
+ & = \frac{vu' - uv'}{uv} \\
+ y' & = y\frac{vu' - uv'}{uv} \\
+ & = \frac{u}{v}\frac{vu' - uv'}{uv} \\
+ & = \frac{vu' - uv'}{v^2} \\
+\end{align}$$
+
+## logarithmic proofs
+
+As well as being used in the proof of the quotient rule, logarithms can also be
+used to prove a couple of other derivative rules.
+
+### 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
+
+$$\begin{align}
+ y & = uv \\
+\mathrm{ln} y & = \mathrm{ln} uv \\
+ & = \mathrm{ln} u + \mathrm{ln} v \\
+ \frac{y'}{y} & = \frac{u'}{u} + \frac{v'}{v} \\
+ y' & = y \frac{u'}{u} + \frac{v'}{v} \\
+ & = uv \left(\frac{u'}{u} + \frac{v'}{v}\right) \\
+ & = vu' + uv' \\
+\end{align}$$
diff --git a/_posts/2017-10-04-quotient-rule.md b/_posts/2017-10-04-quotient-rule.md
@@ -1,43 +0,0 @@
----
-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
-
-$$\begin{align}
- y & = \frac{u}{v} \\
- & = uv^{-1} \\
-y' & = v^{-1}u' + u(-v^{-2}v') \\
- & = \frac{u'}{v} - \frac{uv'}{v^2} \\
- & = \frac{v}{v}\cdot\frac{u'}{v} - \frac{uv'}{v^2} \\
- & = \frac{vu'}{v^2} - \frac{uv'}{v^2} \\
- & = \frac{vu' - uv'}{v^2} \\
-\end{align}$$
