Differentiation - Teach.sg

Differentiation Rules

$\,\frac{\text{d}}{\text{d}x}(c) = 0$

$\,\frac{\text{d}}{\text{d}x}(x^n) = nx^{n-1}$

$\,\frac{\text{d}}{\text{d}x}(\sin x) = \cos x$

$\,\frac{\text{d}}{\text{d}x}(\cos x) = -\sin x$

$\,\frac{\text{d}}{\text{d}x}(\tan x) = \sec^2 x$

$\,\frac{\text{d}}{\text{d}x}(e^{x}) = e^{x}$

$\,\frac{\text{d}}{\text{d}x}(\ln x) = \frac{1}{x}$

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)

$\,\frac{\text{d}}{\text{d}x}[(ax+b)^n] = an(ax+b)^{n-1}$

$\,\frac{\text{d}}{\text{d}x}[\sin(ax+b)] = a\cos(ax+b)$

$\,\frac{\text{d}}{\text{d}x}[\cos(ax+b)] = -a\sin(ax+b)$

$\,\frac{\text{d}}{\text{d}x}[\tan(ax+b)] = a\sec^2(ax+b)$

$\,\frac{\text{d}}{\text{d}x}(e^{ax+b}) = ae^{ax+b}$

$\,\frac{\text{d}}{\text{d}x}[\ln(ax+b)] = \frac{a}{ax+b}$

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 $y=\text{f}(x)$,

$\dfrac{\text{d}y}{\text{d}x} \text{ is the gradient of the curve.}$

Gradient Of Tangent & Normal

From $\dfrac{\text{d}y}{\text{d}x}$, substitute $x=k$ to get $m$ (gradient of tangent).

$-\dfrac{1}{m} \text{ is the gradient of the normal.}$

Increasing & Decreasing Functions

$\text{1. For increasing functions, } \dfrac{\text{d}y}{\text{d}x}>0$

$\text{2. For decreasing functions, } \dfrac{\text{d}y}{\text{d}x}<0$

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

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

Second Derivative Test

Maximum point

$\dfrac{\text{d}^2y}{\text{d}x^2} < 0$

Minimum point

$\dfrac{\text{d}^2y}{\text{d}x^2} > 0$

Inconclusive

$\dfrac{\text{d}^2y}{\text{d}x^2} = 0$

Use first derivative test