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]
⇒ = 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,