Question

What is the derivative of $\sin \left( \sin x \right)$ ?

a)$\cos \left( \cos x \right)$
b)$\cos \left( \sin x \right)$
c)$\cos \left( \sin x \right)\cos x$
d)$\cos \left( \cos x \right)\cos x$

Answer 1

Hint: It is a type of function of function. So, derivative of function of function can be done as first differentiate first differentiate first function keeping the second one as it is. Now differentiate the second function. Now the both values you got keep them together and multiply. This multiplication result is nothing but the differentiation value. $\dfrac{d}{dx}\sin x=\cos x$

Complete step-by-step answer:

Given function in the question of sine x is given as:

$\sin \left( \sin x \right)$

First Step: Differentiate first one keeping the second one as it is.

$\dfrac{d}{dx}\left( \sin \left( \sin x \right) \right)$

The term differentiation of first term is given by equation:

By basic knowledge of the differentiation we can say that:

$\dfrac{d}{dx}\sin x=\cos x$

By differentiating the first function gives us the expression:

$\cos x\left( \sin x \right)$

The second function is kept as it is. So, let us assume this expression to be represented as “A”. (temporarily).

$A=\cos x\left( \sin x \right)$

Second step: Differentiating the second function in the expression.

The second function in the expression is given by the:

$\sin x$

By basic knowledge of trigonometry we can say that the:

$\dfrac{d}{dx}\sin x=\cos x$

So, applying that here we have second function differentiation:

$\dfrac{d}{dx}\sin x$

By substituting above equation, we get second function differentiation as

$\cos x$

Let this expression be denoted by B: $B=\cos x$

We know the product of both the equations is the result.

So, $\dfrac{d}{dx}\left( \sin \left( \sin x \right) \right)=A\cdot B$

By substituting the values of A, B into above; we get

$\dfrac{d}{dx}\left( \sin \left( \sin x \right) \right)=\cos \left( \sin x \right)\cos x$

Hence, $\cos \left( \sin x \right)\cos x$ is required to result in the question.

Option (c) is the correct answer for a given question.

Note: Alternate method is to assume the expression as y. Then apply ${{\sin }^{-1}}$ on both sides, then differentiate to find $\dfrac{dy}{dx}$ . This also at last gives the same answer.