Differentiate with respect to if

Let and v = sin^–1(3x – 4x³)

We need to differentiate u with respect to v that is find.

We have

By substituting x = tan θ, we have

Given,

However, x = tan θ

As tan 0 = 0 and tan = 1, we have .

Thus, lies in the range of tan^–1x.

Hence,

On differentiating u with respect to x, we get

We know and derivative of a constant is 0.

Now, we have v = sin^–1(3x – 4x³)

By substituting x = sin θ, we have

v = sin^–1(3sinθ – 4sin³θ)

But, sin3θ = 3sinθ – 4sin³θ

⇒ v = sin^–1(sin3θ)

Given,

However, x = sin θ

Hence, v = sin^–1(sin3θ) = 3θ

⇒ v = 3sin^–1x

On differentiating v with respect to x, we get

We know

We have

Thus,