--- /dev/null
+---
+geometry: margin=2cm
+<!-- columns: 2 -->
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+---
+
+# Spec - Calculus
+
+## Gradients
+
+$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
+
+## Limit theorems
+
+1. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
+2. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
+3. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
+4. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
+
+
+## First principles derivative
+
+$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
+
+
+## Tangents & gradients
+
+**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$
+**Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$)
+**Secant** $={{f(x+h)-f(x)} \over h}$
+
+## Derivatives
+
+| $f(x)$ | $f^\prime(x)$ |
+| ------ | ------------- |
+| $kx^n$ | $knx^{n-1}$ |
+| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ |
+| $c$ | $0$ |
+| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ |
+| $uv$ | $u{dv \over dx} + v{du \over dx}$ |
+| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ |
+| $\sin ax$ | $a\cos ax$ |
+| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
+| $\cos ax$ | $-a \sin ax$ |
+| $e^{ax}$ | $ae^{ax}$ |
+| $\log_e {ax}$ | $1 \over x$ |
+| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
+
+
+
+## Product rule for $y=uv$
+
+$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
+
+## Logarithms
+
+$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
+
+## Integration
+
+$$\int f(x) dx = F(x) + c$$
+
+- area enclosed by curves
+
+| $f(x)$ | $\int f(x) \cdot dx$ |
+| --------------- | ------------------ |
+| $k$ (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)|
+| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
+| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
+| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
+
+
+## Definite integrals
+
+$$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}$$
+
+## Kinematics
+
+**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
+
+| | no |
+| - | -- |
+| $v=u+at$ | $s$ |
+| $s=ut + {1 \over 2} at^2$ | $v$ |
+| $v^2 = u^2 + 2as$ | $t$ |
+| $s= {1 \over 2}(u+v)t$ | $a$ |