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   1# Differential calculus
   2
   3## Limits
   4
   5$$\lim_{x \rightarrow a}f(x)$$
   6
   7$L^-$ - limit from below
   8
   9$L^+$ - limit from above
  10
  11$\lim_{x \to a} f(x)$ - limit of a point  
  12
  13- Limit exists if $L^-=L^+$
  14- If limit exists, point does not.
  15
  16Limits can be solved using normal techniques (if div 0, factorise)
  17
  18## Limit theorems
  19
  201. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
  212. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
  223. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
  234. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
  24
  25Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant
  26
  27## Solving limits for $x\rightarrow\infty$
  28
  29Factorise so that all values of $x$ are in denominators.
  30
  31e.g.
  32
  33$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$
  34
  35
  36## Continuous functions
  37
  38A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
  39
  40## Gradients of secants and tangents
  41
  42Secant (chord) - line joining two points on curve
  43
  44Tangent - line that intersects curve at one point
  45
  46given $P(x,y) \quad Q(x+\delta x, y + \delta y)$:
  47gradient of chord joining $P$ and $Q$ is ${m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}$
  48
  49As $Q \rightarrow P, \delta x \rightarrow 0$. Chord becomes tangent (two infinitesimal points are equal).
  50
  51Can also be used with functions, where $h=\delta x$.
  52
  53## First principles derivative
  54
  55$$f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}$$
  56
  57$$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$
  58
  59
  60
  61$$m_{\operatorname{chord PQ}}=f^\prime(x)$$
  62
  63first principles derivative:
  64$${m_{\operatorname{tangent at P}} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}$$
  65
  66## Gradient at a point
  67
  68Given point $P(a, b)$ and function $f(x)$, the gradient is $f^\prime(a)$
  69
  70
  71## Derivatives of $x^n$
  72
  73For $f: \mathbb{R} \rightarrow \mathbb{R}$ where $f(x)=x^n, x \in \mathbb{N}$
  74
  75Derivative is $f^\prime(x) = nx^{n-1}$
  76
  77If $x=$ constant, derivative is $0$
  78
  79If $f(x)={1 \over x}=x^{-1}, \quad f^\prime(x)=-1x^{-2}={-1 \over x^2}$
  80
  81If $f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}$
  82
  83If $f(x)=(x-b)^2, \quad f^\prime(x)=2(x-b)$
  84
  85$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  86
  87## Euler's number as a limit
  88
  89$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
  90
  91## Chain rule
  92
  93Leibniz notation:
  94
  95$${dy \over dx} = {dy \over du} \times {du \over dx}$$
  96
  97Function notation:
  98
  99$$(f\circ g)^\prime(x)=f^\prime(g(x))g^\prime(x),\quad \mathbb{where}\hspace{0.3em} (f\circ g)(x)=f(g(x))$$
 100
 101Used with only one expression.
 102
 103e.g. $y=(x^2+5)^7$ - Cannot reasonably expand  
 104Let $u-x^2+5$ (inner expression)  
 105${du \over dx} = 2x$  
 106$y=u^7$  
 107${dy \over du} = 7u^6$  
 108
 109
 110$7u^6 \times$
 111
 112## Product rule
 113
 114If $f(x)=u(x) \times v(x)$, then $f^\prime (x) = u(x) \times v^\prime(x) + v(x)\times u^\prime(x)$
 115
 116If $y=uv$, then derivative ${dy \over dx} = u{dv \over dx} + v{du \over dx}$
 117
 118Surds can be left on denomintaors.
 119
 120## Quotient rule
 121
 122If $f(x)={u(x) \over v(x)}$, then $f^\prime(x)={{v(x)u^\prime(x)-u(x)v^\prime(x)} \over [v(x)]^2}$
 123
 124If $y={u(x) \over v(x)}$, then derivative ${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$
 125