Differentiate with respect to if 0 < x < 1.
Let and.
We need to differentiate u with respect to v that is find.
We have
By substituting x = tan θ, we have
[∵ sec2θ – tan2θ = 1]
⇒ u = sin–1(2sinθcosθ)
But, sin2θ = 2sinθcosθ
⇒ u = sin–1(sin2θ)
Given 0 < x < 1 ⇒ x ϵ (0, 1)
However, x = tan θ
⇒ tan θ ϵ (0, 1)
Hence, u = sin–1(sin2θ) = 2θ
⇒ u = 2tan–1x
On differentiating u with respect to x, we get
We know
Now, we have
By substituting x = tan θ, we have
[∵ sec2θ – tan2θ = 1]
⇒ v = cos–1(cos2θ – sin2θ)
But, cos2θ = cos2θ – sin2θ
⇒ v = cos–1(cos2θ)
However,
Hence, v = cos–1(cos2θ) = 2θ
⇒ v = 2tan–1x
On differentiating v with respect to x, we get
We know
We have
Thus,