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S Chand Class 12 ICSE Maths Solutions Chapter 8 Differentiation Ex 8(c)
Differentiate the following functions w.r.t. x :
Question 1.
sin²(x²)
Solution:
Let y = sin²x² = (sin x²)²
∴ \(\frac{d y}{d x}=\frac{d}{d x}\left(\sin x^2\right)^2=2 \sin x^2 \frac{d}{d x} \sin x^2\) = 2 sin² cosx². 2x
= 2x sin(2x²) [∵ 2 sin θ cos θ = sin² θ]
Question 2.
sin\(\frac{1+x^2}{1-x^2}\)
Solution:
Question 3.
x²sin(\(\frac{1}{x}\))
Solution:
Question 4.
\(\sqrt{a \sin ^2 x+b \cos ^2 x}\)
Solution:
Question 5.
cos (1 – x²)²
Solution:
Let y = cos (1 – x²)²; Diff both sides of eqn (1) w.r.t x ; we have
\(\frac{d y}{d x}=-\sin \left(1-x^2\right)^2 \frac{d}{d x}\left(1-x^2\right)^2=-\sin \left(1-x^2\right)^2 2\left(1-x^2\right) \frac{d}{d x}\left(1-x^2\right)\)
= \(-\sin \left(1-x^2\right)^2 2\left(1-x^2\right)(-2 x)=4 x\left(1-x^2\right)^2 \sin \left(1-x^2\right)^2\)
Question 6.
sin²x cos³ x
Solution:
Question 7.
cot (sin \(\sqrt{x}\))
Solution:
Question 8.
\(\sqrt{\frac{\sec x-1}{\sec x+1}}\)
Solution:
Question 9.
y = \(\frac{\sin x+x^2}{\cot 2 x}\)
Solution:
Diff both sides w.r.t. x ; we have
\(\frac { dy }{ dx }\) = (sinx + x²)\(\frac { d }{ dx }\) tan2x + tan2x \(\frac { d }{ dx }\) (sinx + x²)
= (sinx + x²)sec² 2x.2 + tan2x (cosx + 2x)
= (2sinx + 2x²)sec² 2x + (cosx + 2x) tan2x
Question 10.
If y = \(\left(\frac{\sec x-\tan x}{\sec x+\tan x}\right)\), show that \(\frac{d y}{d x}=-\frac{2 \sec x}{(\sec x+\tan x)^2}\)
Solution:
Question 11.
If f(x) = 9 sin x + sin 3x, find f'(\(\frac { π }{ 3 }\)).
Solution:
Given f(x) = 9 sin x + sin 3x ; Diff both sides w.r.t. x ; we have
f'(x) = 9 cos x + 3 cos 3x
∴ \(f^{\prime}\left(\frac{\pi}{3}\right)=9 \cos \frac{\pi}{3}+3 \cos \pi=\frac{9}{2}+3(-1)=\frac{3}{2}\)
Question 12.
Differentiate y = x tan x and show that x sin² x \(\frac { dy }{ dx }\) = x tanx + y sin² x.
Solution:
Given y = x tan x
Diff eqn (1) both sides w.r.t. x ; we have
\(\frac { dy }{ dx }\) = x sec² x + tan x
⇒ x sin²x \(\frac { dy }{ dx }\) = x sin²x [ x sec² x + tan x] = x² tan²x + sin²x (x tan x)
⇒ x sin²x \(\frac { dy }{ dx }\) = x²tan²x + y sin² x [ using eqn (1)]
Question 13.
If y = a sin x + b cos x, show that y² + \(\left(\frac{d y}{d x}\right)^2=a^2+b^2\)
Solution:
Given y = a sin x + b cos x
Diff eqn (1) both sides w.r.t. x ; we have
\(\frac { dy }{ dx }\) = a cos x – b sin x
on squaring and adding eqn. (1) & eqn. (2) ; we have
y² + (\(\frac { dy }{ dx }\))² = (a sin x + b cos x)² + (a cos x – b sin x)²
= a²(sin² x + cos² x) + b²(sin² x + cos² x) – a² + b²
Question 14.
If y = \(\sqrt{\frac{1-\sin 2 x}{1+\sin 2 x}} \text {, show that } \frac{d y}{d x}+\sec ^2\left(\frac{\pi}{4}-x\right)\) = 0.
Solution:
Question 15.
If y = 2 tan \(\frac { x }{ 2 }\), prove that \(\frac{d y}{d x}=\frac{2}{1+\cos x}\).
Solution:
Given y = 2 tan\(\frac { x }{ 2 }\), Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=2 \sec ^2 \frac{x}{2} \cdot \frac{1}{2}=\frac{1}{\cos ^2 \frac{x}{2}}=\frac{1}{\frac{1+\cos \left(2 \times \frac{x}{2}\right)}{2}}=\frac{2}{1+\cos x}\)