$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
-## Chain rule
+## Chain rule for $(f\circ g)$
$$(f \circ g)^\prime = (f^\prime \circ g) \cdot g^\prime$$
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
+
+$$\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y$$
+$$\ln x = \log_e x$$
+
+### Differentiating logarithms
+$${d(\log_e x)\over dx} = x^-1 = {1 \over x}$$
+
+## Solving $e^x$
+
+| $f(x)$ | $f^\prime(x)$ |
+| ------ | ------------- |
+| $\sin x$ | $\cos x$ |
+| $\sin ax$ | $a\cos ax$ |
+| $\cos x$ | $-\sin x$ |
+| $\cos ax$ | $-a \sin ax$ |
+| $e^x$ | $e^x$ |
+| $e^{ax}$ | $ae^{ax}$ |
+| $\log_e x$ | $1 \over x$ |
+| $\log_e {ax}$ | $1 \over x$ |
+