Differentiate with respect to if
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]
⇒ u = sin–1(2sinθcosθ)
⇒ u = sin–1(sin2θ)
Given
However, x = sin θ
Hence, u = sin–1(sin 2θ) = 2θ.
⇒ u = 2sin–1(x)
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 = tan–1(tanθ)
We have
Hence, v = tan–1(tanθ) = θ
⇒ v = sin–1x
On differentiating v with respect to x, we get
We know
We have
Thus,