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UPSC NDA I 2019 Mathematics Solved Paper

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Question : 79 of 120

Marks: +1, -0

If f(x) = sin (cos x), then f’(x) is equal to

cos (cos x)
sin (- sin x)
(sin x) cos (cos x)
(-sin x) cos (cos x)

Solution:

Concept:

Chain rule of differentiation:

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

.

Also,

\frac{d}{dx} \sin x = \cos x

and

\frac{d}{dx} \cos x = -\sin x

.

Explanation:

Given

f(x) = \sin(\cos x)

.

Differentiate with respect to

x

using the chain rule.

Let

u = \cos x

, then

f = \sin u

.

So

f'(x) = \cos u \cdot \frac{du}{dx}

.

Now

\frac{du}{dx} = -\sin x

.

Therefore

f'(x) = \cos(\cos x) \cdot (-\sin x)

.

This simplifies to

f'(x) = (-\sin x) \cos(\cos x)

.

Answer:

f'(x) = (-\sin x) \cos(\cos x)

, which corresponds to option D.

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