From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sat, 18 Aug 2018 06:55:14 +0000 (+1000)
Subject: logarithmic derivatives
X-Git-Tag: yr11~65
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/e8e9140bc7e4ad3cfdcbc9e0d0877ee9095b226e

logarithmic derivatives
---

e8e9140bc7e4ad3cfdcbc9e0d0877ee9095b226e
diff --cc spec/calculus.md
index 1f3d404,19f97b5..2b68bf5
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@@ -130,25 -130,16 +130,37 @@@ If $f(x)={u(x) \over v(x)}$, then $f^\p
  
  If $y={u(x) \over v(x)}$, then derivative ${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$
  
 +## Logarithms
 +
 +$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
 +
 +Wikipedia:
 +
 +> the logarithm of a given number $x$ is the exponent to which another fixed number, the base $b$, must be raised, to produce that number $x$
 +
 +### Logarithmic identities  
 +$\log_b (xy)=\log_b x + \log_b y$  
 +$\log_b x^n = n \log_b x$  
 +$\log_b y^{x^n} = x^n \log_b y$
 +
 +### $e$ as a logarithm
 +
 +$$\log_e e = 1$$
 +$$\ln x = \log_e x$$
 +
 +### Differentiating logarithms
 +$${d \over dx} \log_b x = {1 \over x \ln b}$$
 +
+ ## Solving $e^x$
+ 
+ | $f(x)$ | $f^\prime(x)$ |
+ | ------ | ------------- |
+ | $\sin x$ | $\cos x$ |
+ | $\sin ax$ | $a\cos ax$ |
+ | $\cos x$ | $-\sin x$ |
+ | $\cos ax$ | $-a \sin ax$ |
+ | $e^x$ | $e^x$ |
+ | $e^{ax}$ | $ae^{ax}$ |
+ | $\log_e x$ | $1 \over x$ |
+ | $\log_e {ax}$ | $1 \over x$ |