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

$\dfrac{\text{d}}{\text{d}x}\left(\dfrac{u}{v}\right)=\dfrac{v\frac{\text{d}u}{\text{d}x}-u\frac{\text{d}v}{\text{d}x}}{v^2}$

Gradient Of Curve

From $y=\text{f}(x)$ ,
$\dfrac{\text{d}y}{\text{d}x}$ is the gradient of the curve.

Gradient Of Tangent & Normal

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 $.[/stextbox] [stextbox caption="Rates Of Change" stextbox id="info"]$ \dfrac{\text{d}y}{\text{d}t}=\dfrac{\text{d}y}{\text{d}x}\times\dfrac{\text{d}x}{\text{d}t} $[/stextbox] [stextbox caption="First Derivative Test" stextbox id="default"]If$ \dfrac{\text{d}y}{\text{d}x}=0 $for$ x=k $, test for$ k^- $,$ k $,$ k^+

$. Maximum point: <img src="https://teach.sg/wp-content/ql-cache/quicklatex.com-9f9378ea29ee6f36752bf951c2f47e82_l3.png" height="61" width="142" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & + & 0 & - \\ \hline \end{array}\]" title="Rendered by QuickLaTeX.com"/> Minimum point: <img src="https://teach.sg/wp-content/ql-cache/quicklatex.com-e34eb65dd85078db31a0864a230b4e33_l3.png" height="61" width="142" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & - & 0 & + \\ \hline \end{array}\]" title="Rendered by QuickLaTeX.com"/> Inflexion point: <img src="https://teach.sg/wp-content/ql-cache/quicklatex.com-6c08639adf6fd54c2707aa0c3de17366_l3.png" height="98" width="142" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[\begin{array}{|c|c|c|c|} \hline x & k^- & k & k^+\\\hline \dfrac{\text{d}y}{\text{d}x} & + & 0 & + \\\hline \dfrac{\text{d}y}{\text{d}x} & - & 0 & - \\\hline \end{array}\]" title="Rendered by QuickLaTeX.com"/>[/stextbox] [stextbox caption="Second Derivative Test" stextbox id="download"]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.