Full of Math Examples
FULL OF MATH EXAMPLES

# Differentiation Examples

Here are the all important examples on differentiation.

## Differentiation Chain Rule

### Example #1

#### Derivative of $\sin(x^{2})$.

$y=\sin \left(x^{2}\right) \Rightarrow \frac{d y}{d x}=\frac{d}{d x}\left(\sin x^{2}\right)$
$=\cos x^{2} \frac{d}{d x}\left(x^{2}\right)=2 x \cos x^{2}$

### Example #2

#### Derivative of $\sin \left(\cos x^{2}\right)$.

$y=\sin \left(\cos x^{2}\right)$
$\frac{d y}{d x}=\frac{d}{d x} \sin \left(\cos x^{2}\right)$$=\cos \left(\cos x^{2}\right) \frac{d}{d x}\left(\cos x^{2}\right) =\cos \left(\cos x^{2}\right)\left(-\sin x^{2}\right) \frac{d}{d x}\left(x^{2}\right) =-\sin x^{2} \cos \left(\cos x^{2}\right)(2 x) =-2 x \sin x^{2} \cos \left(\cos x^{2}\right) ## Differentiation Product Rule ### Example #3 #### Derivative of f(x)=\left(6 x^{2}+2 x\right)\left(x^{3}+1\right) . Let u(x)=6 x^{2}+2 x and v(x)=x^{3}+1 . Then \frac{d u}{d x}=12 x+2 \frac{d v}{d x}=3 x^{2} from the product rule, \frac{d f}{d x}=u \times \frac{d v}{d x}+v \times \frac{d u}{d x} Then, \frac{d f}{d x}=\left(6 x^{2}+2 x\right)\left(3 x^{2}\right)+\left(x^{3}+1\right)(12 x+2) =18 x^{4}+6 x^{3}+12 x^{4}+2 x^{3}+12 x+2 =30 x^{4}+8 x^{3}+12 x+2 ### Example #4 #### Derivative of x^{3} \sin x . \frac{d}{d x}\left(x^{3} \sin x\right)$$=x^{3} \frac{d}{d x}(\sin x)+\frac{d}{d x}\left(x^{3}\right) \sin x$
$=x^{3} \cos x+3 x^{2} \sin x$

## Differentiation of Quotients

$f^{\prime}=\left(\frac{u}{v}\right)^{\prime}=\frac{v u^{\prime}-u v^{\prime}}{v^{2}}$

### Example #5

#### Derivative of $\sin x / x$.

From formula we have $u(x)=\sin x, v(x)=x$ and hence $u^{\prime}(x)=\cos x, v^{\prime}(x)=1,$ Then,
$\frac{d}{d x}\left(\frac{\sin x}{x}\right)$$=\frac{x \cos x-\sin x}{x^{2}}=\frac{\cos x}{x}-\frac{\sin x}{x^{2}}$

### Example #6

#### Derivative of $f(x)=\frac{x^{2}+7}{3 x-1}$.

Here, $u(x)=x^{2}+7$ and $v(x)=3 x-1$
Then, $\frac{d u}{d x}=2 x$ and $\frac{d v}{d x}=3$
Using quotient rule,
$\frac{d f}{d x}=\frac{(3 x-1)(2 x)-\left(x^{2}+7\right)(3)}{(3 x-1)^{2}}$
$=\frac{6 x^{2}-2 x-3 x^{2}-21}{(3 x-1)^{2}}$

## Logarithmic Differentiation

### Example #7

#### Differentiation of $a^{x}$.

Let $y=a^{x}$ .Then,
$\log y=x \log a$
Differentiating both side with respect to x gives
$\frac{1}{y} \frac{d y}{d x}=\log a$
$\frac{d y}{d x}=y \log a$
Thus, $\frac{d}{d x}\left(a^{x}\right)=a^{x} \log a$

### Example #8

#### Differentiation of $(\cos x)^{\sin x}$.

Taking logarithm of both side and differentiating,
$f(x)=(\cos x)^{\sin x}$
$\ln f(x)=\sin x \cdot \ln (\cos x)$
$\frac{f^{\prime}(x)}{f(x)}=\cos x \ln (\cos x)+\sin x \cdot \frac{-\sin x}{\cos x}$
$=\cos x \ln (\cos x)-\frac{\sin ^{2} x}{\cos x}$
$f^{\prime}(x)=f(x)\left(\cos x \ln (\cos x)-\frac{\sin ^{2} x}{\cos x}\right)$
$f^{\prime}(x)=(\cos x)^{\sin x+1} \ln (\cos x)-(\cos x)^{\sin x-1} \sin ^{2} x$