Differentiation

Differentiation Rules
$1.\,\frac{\text{d}}{\text{d}x} c = 0$
$2.\,\frac{\text{d}}{\text{d}x} x^n = nx^{n-1}$
$3.\,\frac{\text{d}}{\text{d}x} \sin x = \cos x$
$4.\,\frac{\text{d}}{\text{d}x} \cos x = -\sin x$
$5.\,\frac{\text{d}}{\text{d}x} \tan x = \sec^2 x$
$6.\,\frac{\text{d}}{\text{d}x} e^{x} = e^{x}$
$7.\,\frac{\text{d}}{\text{d}x} \ln\; x = \frac{1}{x}$
Note: $\frac{\text{d}}{\text{d}x} k\text{f}(x)=k\times\frac{\text{d}}{\text{d}x} \text{f}(x)$
Chain Rule
For $y=\text{f}(u)$ and $u=\text{g}(x)$,
$\dfrac{\text{d}y}{\text{d}x} = \dfrac{\text{d}y}{\text{d}u} \times \dfrac{\text{d}u}{\text{d}x}$
Further Differentiation Rules (Chain Rule)
$1.\,\frac{\text{d}}{\text{d}x} (ax+b)^n = an(ax+b)^{n-1}$
$2.\,\frac{\text{d}}{\text{d}x} \sin (ax+b) = a\cos (ax+b)$
$3.\,\frac{\text{d}}{\text{d}x} \cos (ax+b) = -a\sin (ax+b)$
$4.\,\frac{\text{d}}{\text{d}x} \tan (ax+b) = a\sec^2 (ax+b)$
$5.\,\frac{\text{d}}{\text{d}x} e^{ax+b} = ae^{ax+b}$
$6.\,\frac{\text{d}}{\text{d}x} \ln\; (ax+b) = \frac{a}{ax+b}$
Product Rule
$\dfrac{\text{d}}{\text{d}x}(uv) = u\dfrac{\text{d}v}{\text{d}x} + v\dfrac{\text{d}u}{\text{d}x}$
Quotient Rule
$\dfrac{\text{d}}{\text{d}x}\left(\dfrac{u}{v}\right)=\dfrac{v\frac{\text{d}u}{\text{d}x}-u\frac{\text{d}v}{\text{d}x}}{v^2}$
From $y=\text{f}(x)$,
$\dfrac{\text{d}y}{\text{d}x}$ is the gradient of the curve.
From $\dfrac{\text{d}y}{\text{d}x}$, substitute $x=k$ to get $m$ (gradient of tangent).
$-\dfrac{1}{m}$ is the gradient of the normal.
Increasing & Decreasing Functions
1. For increasing functions, $\dfrac{\text{d}y}{\text{d}x}>0\\$.
2. For decreasing functions, $\dfrac{\text{d}y}{\text{d}x}<0$.
Rates Of Change
$\dfrac{\text{d}y}{\text{d}t}=\dfrac{\text{d}y}{\text{d}x}\times\dfrac{\text{d}x}{\text{d}t}$
First Derivative Test
If $\dfrac{\text{d}y}{\text{d}x}=0$ for $x=k$, test for $k^-$, $k$, $k^+$.

Maximum point:
$$\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & + & 0 & – \\ \hline \end{array}$$

Minimum point:
$$\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & – & 0 & + \\ \hline \end{array}$$

Inflexion point:
$$\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & + & 0 & + \\\hline \dfrac{\text{d}y}{\text{d}x} & – & 0 & – \\\hline \end{array}$$

Second Derivative Test
1. If $\dfrac{\text{d}^2y}{\text{d}x^2}<0\\$, it is a maximum point.
2. If $\dfrac{\text{d}^2y}{\text{d}x^2}>0$, it is a minimum point.
3. If $\dfrac{\text{d}^2y}{\text{d}x^2}=0$, need to do first derivative test.
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