$L^+$ - limit from above
-$\lim_{x \to a} f(x)$ - limit of a point
+$\lim_{x \to a} f(x)$ - limit of a point
- Limit exists if $L^-=L^+$
- If limit exists, point does not.
## First principles derivative
-$$\lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx} = f^\prime(x)$$
+$$f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}$$
$$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$
$$m_{\operatorname{chord PQ}}=f^\prime(x)$$
first principles derivative:
-$${m_{\operatorname{tangent at P}} =\lim_{h \rigzhtarrow 0}}{{f(x+h)-f(x)}\over h}$$
+$${m_{\operatorname{tangent at P}} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}$$
+## Gradient at a point
+Given point $P(a, b)$ and function $f(x)$, the gradient is $f^\prime(a)$
+## Derivatives of $x^n$
+
+$${d(ax^n) \over dx}=anx^{n-1}$$
+
+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 $f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}$
+
+If $f(x)=(x-b)^2, \quad f^\prime(x)=2(x-b)$
+
+$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
+
+## 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 for $(f\circ g)$
+
+$$(f \circ g)^\prime = (f^\prime \circ g) \cdot g^\prime$$
+
+Leibniz notation:
+
+$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
+
+Function notation:
+
+$$(f\circ g)^\prime(x)=f^\prime(g(x))g^\prime(x),\quad \mathbb{where}\hspace{0.3em} (f\circ g)(x)=f(g(x))$$
+
+Used with only one expression.
+
+e.g. $y=(x^2+5)^7$ - Cannot reasonably expand
+Let $u-x^2+5$ (inner expression)
+${du \over dx} = 2x$
+$y=u^7$
+${dy \over du} = 7u^6$
+
+
+$7u^6 \times$
+
+## Product rule for $y=uv$
+
+$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
+
+Surds can be left on denomintaors.
+
+## 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 $e^x$
+
+| $f(x)$ | $f^\prime(x)$ |
+| ------ | ------------- |
+| $\sin x$ | $\cos x$ |
+| $\sin ax$ | $a\cos ax$ |
+| $\cos x$ | $-\sin x$ |
+| $\cos ax$ | $-a \sin ax$ |
+| $e^x$ | $e^x$ |
+| $e^{ax}$ | $ae^{ax}$ |
+| $\log_e x$ | $1 \over x$ |
+| $\log_e {ax}$ | $1 \over x$ |
+