+---
+geometry: margin=2cm
+columns: 2
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
+---
+
# Differential calculus
## Limits
$L^+$ - 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.
$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$
-
## Continuous functions
A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
+
+## Gradients of secants and tangents
+
+Secant (chord) - line joining two points on curve
+
+Tangent - line that intersects curve at one point
+
+given $P(x,y) \quad Q(x+\delta x, y + \delta y)$:
+gradient of chord joining $P$ and $Q$ is ${m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}$
+
+As $Q \rightarrow P, \delta x \rightarrow 0$. Chord becomes tangent (two infinitesimal points are equal).
+
+Can also be used with functions, where $h=\delta x$.
+
+## First principles derivative
+
+$$f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}$$
+
+$$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$
+
+
+
+$$m_{\operatorname{chord PQ}}=f^\prime(x)$$
+
+first principles derivative:
+$${m_{\operatorname{tangent at P}} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}$$
+
+## 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 $(f\circ g)$
+
+If $f(x) = h(g(x)) = (h \circ g)(x)$:
+
+$$f^\prime(x) = h^\prime(g(x)) \cdot g^\prime(x)$$
+
+If $y=h(u)$ and $u=g(x)$:
+
+$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
+$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
+
+Used with only one expression.
+
+e.g. $y=(x^2+5)^7$ - Cannot reasonably expand
+Let $u-x^2+5$ (inner expression)
+${du \over dx} = 2x$
+$y=u^7$
+${dy \over du} = 7u^6$
+
+## 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 ${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$
+
+## Logarithms
+
+$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
+
+Wikipedia:
+
+> the logarithm of a given number $x$ is the exponent to which another fixed number, the base $b$, must be raised, to produce that number $x$
+
+### Logarithmic identities
+
+$\log_b (xy)=\log_b x + \log_b y$
+$\log_b x^n = n \log_b x$
+$\log_b y^{x^n} = x^n \log_b y$
+
+### 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
+
+$$\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y$$
+$$\ln x = \log_e x$$
+
+### Differentiating logarithms
+$${d(\log_e x)\over dx} = x^{-1} = {1 \over x}$$
+
+## Derivative rules
+
+| $f(x)$ | $f^\prime(x)$ |xs
+| ------ | ------------- |
+| $\sin x$ | $\cos x$ |
+| $\sin ax$ | $a\cos ax$ |
+| $\cos x$ | $-\sin x$ |
+| $\cos ax$ | $-a \sin ax$ |
+| $\tan f(x)$ | $f^2(x) \sec^2f(x)$ |
+| $e^x$ | $e^x$ |
+| $e^{ax}$ | $ae^{ax}$ |
+| $ax^{nx}$ | $an \cdot e^{nx}$ |
+| $\log_e x$ | $1 \over x$ |
+| $\log_e {ax}$ | $1 \over x$ |
+| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
+| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
+| $\sin^{-1} x$ | $1 \over {\sqrt{1-x^2}}$ |
+| $\cos^{-1} x$ | $-1 \over {sqrt{1-x^2}}$ |
+| $\tan^{-1} x$ | $1 \over {1 + x^2}$ |
+
+<!-- $${d(ax^{nx}) \over dx} = an \cdot e^nx$$ -->
+
+Reciprocal derivatives:
+
+$${{dy \over dx} \over 1} = dx \over dy$$
+
+## Differentiating $x=f(y)$
+
+Find $dx \over dy$. Then $dx \over dy = {1 \over {dy \over dx}} \therefore {dy \over dx} = {1 \over {dx \over dy}}$.
+
+$${dy \over dx} = {1 \over {dx \over dy}}$$
+
+## Second derivative
+
+$$f(x) \implies f^\prime (x) \implies f^{\prime\prime}(x)$$
+
+$$\therefore y \implies {dy \over dx} \implies {d({dy \over dx}) \over dx} \implies {d^2 y \over dx^2}$$
+
+Order of polynomial $n$th derivative decrements each time the derivative is taken
+
+### Points of Inflection
+
+*Stationary point* - point of zero gradient (i.e. $f^\prime(x)=0$)
+*Point of inflection* - point of maximum $|$gradient$|$ (i.e. $f^{\prime\prime} = 0$)
+
+- if $f^\prime (a) = 0$ and $f^{\prime\prime}(a) > 0$, then point $(a, f(a))$ is a local min (curve is concave up)
+- if $f^\prime (a) = 0$ and $f^{\prime\prime} (a) < 0$, then point $(a, f(a))$ is local max (curve is concave down)
+- if $f^{\prime\prime}(a) = 0$, then point $(a, f(a))$ is a point of inflection
+- - if also $f^\prime(a)=0$, then it is a stationary point of inflection
+
+
+
+## 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) | $kx + c$ |
+| $x^n$ | ${x^{n+1} \over {n+1}} + c$ |
+| $a x^{-n}$ | $a \cdot \log_e x + c$ |
+| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
+| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
+| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
+| $e^k$ | $e^kx + c$ |
+| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
+| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
+| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
+| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
+| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
+
+## 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.
+
+## Rates
+
+### Related rates
+
+$${da \over db} \quad \text{change in } a \text{ with respect to } b$$
+
+#### Gradient at a point on parametric curve
+
+$${dy \over dx} = {{dy \over dt} \over {dx \over dt}} \> \vert \> {dx \over dt} \ne 0$$
+
+$${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \over {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$
+
+# Rational functions
+
+$$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}$$
+
+## Addition of ordinates
+
+- when two graphs have the same ordinate, $y$-coordinate is double the ordinate
+- when two graphs have opposite ordinates, $y$-coordinate is 0 i.e. ($x$-intercept)
+- when one of the ordinates is 0, the resulting ordinate is equal to the other ordinate
+