Q. 414.5( 2 Votes )

# Differentiate <sp

Let and.

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

We have

By substituting x = sin θ, we have

[ sin2θ + cos2θ = 1]

= tan–1(tanθ)

Given

However, x = sin θ

Hence, u = tan–1(tanθ) = θ

u = sin–1x

On differentiating u with respect to x, we get

We know

Now, we have

By substituting x = sin θ, we have

[ sin2θ + cos2θ = 1]

v = sin–1(2sinθcosθ)

v = sin–1(sin2θ)

However,

Hence, v = sin–1(sin 2θ) = 2θ.

v = 2sin–1(x)

On differentiating v with respect to x, we get

We know

We have

Thus,

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