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