## Chain rule for $(f\circ g)$
-$$(f \circ g)^\prime = (f^\prime \circ g) \cdot g^\prime$$
-
-Leibniz notation:
-
$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
+$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
Function notation:
$\log_b x^n = n \log_b x$
$\log_b y^{x^n} = x^n \log_b y$
+### Index identities
+$b^{m+n}=b^m \cdot b^n$
+$(b^m)^n=b^{m \cdot n}$
+$(b \cdot c)^n = b^n \cdot c^n$
+${a^m \div a^n} = {a^{m-n}}$
+
### $e$ 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 $e^x$
+## Solving $e^x$ etc
-| $f(x)$ | $f^\prime(x)$ |
+| $f(x)$ | $f^\prime(x)$ |xs
| ------ | ------------- |
| $\sin x$ | $\cos x$ |
| $\sin ax$ | $a\cos ax$ |
| $\cos ax$ | $-a \sin ax$ |
| $e^x$ | $e^x$ |
| $e^{ax}$ | $ae^{ax}$ |
+| $ax^{nx}$ | $an \cdot e^{nx}$ |
| $\log_e x$ | $1 \over x$ |
| $\log_e {ax}$ | $1 \over x$ |
+| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
+| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
+
+<!-- $${d(ax^{nx}) \over dx} = an \cdot e^nx$$ -->
+
+## Antidifferentiation
+
+$$y={x^{n+1} \over n+1} + c$$
+
+## Integration
+
+$$\int f(x) dx = F(x) + c$$
+
+- area enclosed by curves
+- $+c$ should be shown on each step without $\int$
+
+$$\int x^n = {x^{n+1} \over n+1} + c$$
+
+### Integral laws
+
+$\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$
+$\int k f(x) dx = k \int f(x) dx$
+
+| $f(x)$ | $\int f(x) \cdot dx$ |
+| ------------------------------- | ---------------------------- |
+| $k$ (constant) | $kx + c$ |
+| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
+| $a x^{-n}$ | $a \cdot \log_e x + c$ |
+| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
+| $e^k$ | $e^kx + c$ |
+| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
+| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
+| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
+| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
+| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
+| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
+| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
+
+## Applications of antidifferentiation
+
+- $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
+- the nature of any stationary point of $y=F(x)$ is determined by the way the sign of the graph of $y=f(x)$ changes about its $x$-intercepts
+- if $f(x)$ is a polynomial of degree $n$, then $F(x)$ has degree $n+1$
+
+To find stationary points of a function, substitute $x$ value of given point into derivative. Solve for ${dy \over dx}=0$. Integrate to find original function.
+
+## Kinematics
+
+$${dV \over dt} = {\operatorname{change in volume} \over \operatorname{respect to time}}$$
+
+**position $x$** - distance from origin or fixed point
+**displacement $s$** - change in position from starting point (vector)
+**velocity $v$** - change in position with respect to time
+**acceleration $a$** - change in velocity
+**speed** - magnitude of velocity
+
+$$v_{\operatorname{avg}}={\Delta x \over \Delta t}={{x_2 - x_1} \over {t_2 - t_1}}$$
+$$\operatorname{speed}_{\operatorname{avg}}={\Delta v \over \Delta t}$$