KC Sinha Mathematics Solution Class 12 Chapter 11 अवकलन (Differentiation) Exercise 11.1 (Q13-Q16)

Exercise 11.1

Question 13

sin²(3x+4)
Sol :
Let y=sin²(3x+4)

Differentiating with respect to x

$\frac{d y}{dx}=\frac{d\left[\sin^{2}(3 x+4)\right]}{d[\sin (3 x+4)]} \times \frac{d[\sin(3 x+4)]}{d(3 x+4)}\times\frac{d(3x+4)}{dx}$

=2.sin(3x+4).cos(3x+4).3

=3sin2(3x+4)

Question 14

$\sec ^{3}\left(\frac{x}{2}\right)$
Sol :
Let y=$\sec ^{3}\left(\frac{x}{2}\right)$

Differentiating with respect to x

$\frac{d y}{dx}=\frac{d\left[\sec ^{3}\left(\frac{x}{2}\right)\right]}{d\left[\sec \left(\frac{3}{2}\right)\right]} \times \frac{d\left(\operatorname{sec} \frac{x}{2}\right)}{d\left(\frac{x}{2}\right)} \times \frac{d\left(\frac{x}{2}\right)}{dx}$

$=3\sin^{2}\left(\frac{x}{2}\right) \cdot \sec \frac{x}{2} \cdot \tan \frac{x}{2} \times \frac{1}{2}$

$=\frac{3}{2} \sec ^{3}\left(\frac{x}{2}\right) \cdot \tan \frac{x}{2}$

Question 15

$\sin \{\cos (\tan \sqrt{x})\}$
Sol :
Let y=$\sin \{\cos (\tan \sqrt{x})\}$

Differentiating with respect to x

$\frac{d y}{d x}=\frac{d[\sin \{\cos (\tan \sqrt{x})\}]}{d[\cos (\tan \sqrt{2})]} \times d \frac{[\cos (\tan \sqrt{2})]}{d(\tan \sqrt{x})} \frac{d(\tan \sqrt{2})}{d(\sqrt{x})}\times \frac{d(\sqrt x)}{dx}$

=cos{cos(tan√x)}{-sin(tan√x)}.sec²((√x)).$\frac{1}{2 \sqrt{x}}$

$=-\dfrac{1}{2 \sqrt{x}} \sec ^{2} \sqrt{x} \sin (\tan \sqrt{x}) \cdot \cos \left[\cos(\tan\sqrt{x})\right]$

Question 16

sin[cos{tan(cot x)}]
Sol :
Let y=sin[cos{tan(cot x)}]

Differentiating with respect to x

$\frac{d y}{d x}=\frac{d[\sin (\cos)(\tan (\cot x)]\}]}{d[\cos \{\tan (\cot x)\}]}+\frac{d[\cos \{\tan (\cot x)\}]}{d\left[\tan \left(\cot x\right)\right]}\times \frac{d[\tan(\cot x)]}{d(\cot x)}\times \frac{d(\cot x)}{dx}$

=cos[cos{tan(cot x)}].[-sin{tan(cot x)}].sec²(cot x).(-cosec²x)

=cosec²x.sec²(cot x).sin{tan(cot x)}.cos[cos{tan(cot x)}]