c59c4b77e306241793268f6d18b4bd26c6efdf7e
   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
  73$${d(ax^n) \over dx}=anx^{n-1}$$
  74
  75If $x=$ constant, derivative is $0$
  76
  77If $y=ax^n$, derivative is $a\times nx^{n-1}$
  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## Derivatives of $u \pm v$
  88
  89$${dy \over dx}={du \over dx} \pm {dv \over dx}$$
  90where $u$ and $v$ are functions of $x$
  91
  92## Euler's number as a limit
  93
  94$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
  95
  96## Chain rule for $(f\circ g)$
  97
  98$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
  99$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
 100
 101Function notation:
 102
 103$$(f\circ g)^\prime(x)=f^\prime(g(x))g^\prime(x),\quad \mathbb{where}\hspace{0.3em} (f\circ g)(x)=f(g(x))$$
 104
 105Used with only one expression.
 106
 107e.g. $y=(x^2+5)^7$ - Cannot reasonably expand  
 108Let $u-x^2+5$ (inner expression)  
 109${du \over dx} = 2x$  
 110$y=u^7$  
 111${dy \over du} = 7u^6$  
 112
 113
 114## Product rule for $y=uv$
 115
 116$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
 117
 118Surds can be left on denomintaors.
 119
 120## Quotient rule for $y={u \over v}$
 121
 122$${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$$
 123
 124If $f(x)={u(x) \over v(x)}$, then $f^\prime(x)={{v(x)u^\prime(x)-u(x)v^\prime(x)} \over [v(x)]^2}$
 125
 126If $y={u(x) \over v(x)}$, then derivative ${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$
 127
 128## Logarithms
 129
 130$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
 131
 132Wikipedia:
 133
 134> 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 $x$
 135
 136### Logarithmic identities
 137
 138$\log_b (xy)=\log_b x + \log_b y$  
 139$\log_b x^n = n \log_b x$  
 140$\log_b y^{x^n} = x^n \log_b y$
 141
 142### Index identities
 143
 144$b^{m+n}=b^m \cdot b^n$  
 145$(b^m)^n=b^{m \cdot n}$  
 146$(b \cdot c)^n = b^n \cdot c^n$  
 147${a^m \div a^n} = {a^{m-n}}$
 148
 149### $e$ as a logarithm
 150
 151$$\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y$$
 152$$\ln x = \log_e x$$
 153
 154### Differentiating logarithms
 155$${d(\log_e x)\over dx} = x^{-1} = {1 \over x}$$
 156
 157## Derivative rules
 158
 159| $f(x)$ | $f^\prime(x)$ |xs
 160| ------ | ------------- |
 161| $\sin x$ | $\cos x$ |
 162| $\sin ax$ | $a\cos ax$ |
 163| $\cos x$ | $-\sin x$ |
 164| $\cos ax$ | $-a \sin ax$ |
 165| $\tan f(x)$ | $f^2(x) \sec^2f(x)$ |
 166| $e^x$ | $e^x$ |
 167| $e^{ax}$ | $ae^{ax}$ |
 168| $ax^{nx}$ | $an \cdot e^{nx}$ |
 169| $\log_e x$ | $1 \over x$ |
 170| $\log_e {ax}$ | $1 \over x$ |
 171| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
 172| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
 173| $\sin^{-1} x$ | $1 \over {\sqrt{1-x^2}}$ |
 174| $\cos^{-1} x$ | $-1 \over {sqrt{1-x^2}}$ |
 175| $\tan^{-1} x$ | $1 \over {1 + x^2}$ |
 176
 177<!-- $${d(ax^{nx}) \over dx} = an \cdot e^nx$$ -->
 178
 179Reciprocal derivatives:
 180
 181$${{dy \over dx} \over 1} = dx \over dy$$
 182
 183## Differentiating $x=f(y)$
 184
 185Find $dx \over dy$. Then $dx \over dy = {1 \over {dy \over dx}} \therefore {dy \over dx} = {1 \over {dx \over dy}}$.
 186
 187$${dy \over dx} = {1 \over {dx \over dy}}$$
 188
 189## Second derivative
 190
 191$$f(x) \implies f^\prime (x) \implies f^{\prime\prime}(x)$$
 192
 193$$\therefore y \implies {dy \over dx} \implies {d({dy \over dx}) \over dx} \implies {d^2 y \over dx^2}$$
 194
 195Order of polynomial $n$th derivative decrements each time the derivative is taken
 196
 197### Maxima and minima
 198
 199- if $f^\prime (a) = 0$ and $f^{\prime\prime}(a) > 0$, then point $(a, f(a))$ is a local min (curve is concave up)
 200
 201- if $f^\prime (a) = 0$ and $f^{\prime\prime} (a) < 0$, then point $(a, f(a))$ is local max (curve is concave down)
 202- if $f^{\prime\prime}(a) = 0$, then point $(a, f(a))$ is a point of inflection
 203- - if also $f^\prime(a)=0$, then it is a stationary point of inflection
 204
 205*Point of inflection* - point of maximum gradient (either +ve or -ve)
 206
 207## Antidifferentiation
 208
 209$$y={x^{n+1} \over n+1} + c$$
 210
 211## Integration
 212
 213$$\int f(x) dx = F(x) + c$$
 214
 215- area enclosed by curves
 216- $+c$ should be shown on each step without $\int$
 217
 218$$\int x^n = {x^{n+1} \over n+1} + c$$
 219
 220### Integral laws
 221
 222$\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$  
 223$\int k f(x) dx = k \int f(x) dx$  
 224
 225| $f(x)$                          | $\int f(x) \cdot dx$         |
 226| ------------------------------- | ---------------------------- |
 227| $k$ (constant) | $kx + c$ |
 228| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
 229| $a x^{-n}$ | $a \cdot \log_e x + c$ |
 230| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
 231| $e^k$ | $e^kx + c$ |
 232| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
 233| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
 234| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
 235| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
 236| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
 237| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
 238| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
 239
 240## Applications of antidifferentiation
 241
 242- $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
 243- 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
 244- if $f(x)$ is a polynomial of degree $n$, then $F(x)$ has degree $n+1$
 245
 246To 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.
 247
 248## Rates
 249
 250### Related rates
 251
 252$${da \over db} \quad \text{change in } a \text{ with respect to } b$$
 253
 254#### Gradient at a point on parametric curve
 255
 256$${dy \over dx} = {{dy \over dt} \over {dx \over dt}} \> \vert \> {dx \over dt} \ne 0$$
 257
 258$${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \over {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$
 259