KC Sinha Mathematics Solution Class 12 Chapter 11 अवकलन (Differentiation) Exercise 11.1 (Q1-Q4)

Exercise 11.1

Question 1

x के सापेक्ष के फलन अर्थात् संयुक्त फलनों के अवकलन पर आधारित प्रश्नः

[Differentiate the following functions with respect to x]

(i) sin(ax+b)
Sol :
Let y= sin(ax+b)

Differentiating with respect to x

$\frac{d y}{d x}=\frac{d[\sin (a x+b)]}{d\left(a{x}+b\right)} \times \frac{d\left(a{x}+b\right)}{d{x}}$

= cos(ax+b)a

=acos(ax+b)

(ii) sin x²
Sol :
Let y=sin x²

Differentiating with respect to x

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

=cos x²2x

=2x.cos x²


(iii) tan(5x+9)
Sol :
Let y=tan(5x+9)

Differentiating with respect to x

$\frac{d y}{d x}=\frac{d[\tan (5 x+1)]}{d(5 x+9)} \times \frac{d(5 x+9)}{d{x}}$

=sec²(5x+9)5

=5sec²(5x+9)


(iv) cos(sin x²)
Sol :
Let y=cos(sin x²)

Differentiating with respect to x

$\frac{d y}{dx}=\frac{\left.d [ \cos \left(\tan x^{2}\right)\right]}{d\left(\sin x^{2}\right)} \times \frac{d\left(\sin x^{2}\right)}{d\left(x^{2}\right)} \times \frac{d\left(x^{2}\right)}{dx}$

=-sin(sin x²) .cos x².2x

=-2x.sin(sin x²) .cos x²


(v) sin³x
Sol :
Let y=sin³x

Differentiating with respect to x

$\frac{d y}{d x}=\frac{d\left(\sin ^{3} x\right)}{d(\sin x)}=\frac{d(\sin x)}{d x}$

=3.sin²x.cosx


(vi) $\sqrt{x^{2}+x+1}$
Sol :
Let y=$\sqrt{x^{2}+x+1}$

Differentiating with respect to x

$\frac{d y}{dx}=\frac{d(\sqrt{x^{2}+x+1})}{d\left(x^{2}+x+1\right)} \times \frac{d\left(x^{2}+x+1\right)}{dx}$

$=\frac{1}{2 \sqrt{x^{2}+x+1}} \times(2 x+1)$

$=\frac{2 x+1}{2 \sqrt{x^{2}+x+1}}$



Differentiate the following functions with respect to x

Question 2

tan(xⁿ)
Sol :
Let y=tan(xⁿ)

Differentiating with respect to x

$\frac{d y}{d x}=\frac{d\left[\tan \left(x^{n}\right)\right]}{d\left(x^{n}\right)} \times \frac{d\left(x^{n}\right)}{d x}$

=sec²(xⁿ).nx^n-1

=nx^n-1.sec²(xⁿ)

Question 3

cosec(cosec x)
Sol :
Let y=cosec(cosec x)

Differentiating with respect to x

$\frac{d{y}}{dx}=\frac{d[\operatorname{cosec}(\operatorname{cosec} x)]}{d[\operatorname{cosec} x)} \times \frac{d(\cos x)}{dx}$

=-cosec(cosec x).cot(cosec x).(-coec x . cot x)

=cosec(cosec x).cot(cosec x).cosec x. cot x

Question 4

tan(x²+3)
Sol :
Let y=tan(x²+3)

Differentiating with respect to x

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

=sec²(x²+3).2x

=2x.sec²(x²+3)