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 $