Differentiate with respect to if –1<x<1, x ≠ 0.
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]
But, cos2θ = 1 – 2sin2θ and sin2θ = 2sinθcosθ.
Given –1 < x < 1 ⇒ x ϵ (–1, 1)
However, x = tan θ
⇒ tan θ ϵ (–1, 1)
Hence,
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 = sin–1(2sinθcosθ)
But, sin2θ = 2sinθcosθ
⇒ v = sin–1(sin2θ)
However,
Hence, v = sin–1(sin2θ) = 2θ
⇒ v = 2tan–1x
On differentiating v with respect to x, we get
We know
We have
Thus,