## Derivatives of $x^n$
-For $f: \mathbb{R} \rightarrow \mathbb{R}$ where $f(x)=x^n, x \in \mathbb{N}$
-
-Derivative is $f^\prime(x) = nx^{n-1}$
+$${d(ax^n) \over dx}=anx^{n-1}$$
If $x=$ constant, derivative is $0$
+If $y=ax^n$, derivative is $a\times nx^{n-1}$
+
If $f(x)={1 \over x}=x^{-1}, \quad f^\prime(x)=-1x^{-2}={-1 \over x^2}$
If $f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}$
$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
+## Derivatives of $u \pm v$
+
+$${dy \over dx}={du \over dx} \pm {dv \over dx}$$
+where $u$ and $v$ are functions of $x$
+
## Euler's number as a limit
$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
-## Chain rule
+## Chain rule for $(f\circ g)$
+
+$$(f \circ g)^\prime = (f^\prime \circ g) \cdot g^\prime$$
Leibniz notation:
-$${dy \over dx} = {dy \over du} \times {du \over dx}$$
+$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
Function notation:
$7u^6 \times$
-## Product rule
-
-If $f(x)=u(x) \times v(x)$, then $f^\prime (x) = u(x) \times v^\prime(x) + v(x)\times u^\prime(x)$
+## Product rule for $y=uv$
-If $y=uv$, then derivative ${dy \over dx} = u{dv \over dx} + v{du \over dx}$
+$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
Surds can be left on denomintaors.
-## Quotient rule
+## Quotient rule for $y={u \over v}$
+
+$${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$$
If $f(x)={u(x) \over v(x)}$, then $f^\prime(x)={{v(x)u^\prime(x)-u(x)v^\prime(x)} \over [v(x)]^2}$
If $y={u(x) \over v(x)}$, then derivative ${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$
+## Solving $e^x$
+
+| $f(x)$ | $f^\prime(x)$ |
+| ------ | ------------- |
+| $\sin x$ | $\cos x$ |
+| $\sin ax$ | $a\cos ax$ |
+| $\cos x$ | $-\sin x$ |
+| $\cos ax$ | $-a \sin ax$ |
+| $e^x$ | $e^x$ |
+| $e^{ax}$ | $ae^{ax}$ |
+| $\log_e x$ | $1 \over x$ |
+| $\log_e {ax}$ | $1 \over x$ |
+