Differentiate with respect to if
Let and.
We need to differentiate u with respect to v that is find.
We have
By substituting 2x = cos θ, we have
[∵ sin2θ + cos2θ = 1]
⇒ u = sin–1(2 cos θ sin θ)
⇒ u = sin–1(sin2θ)
Given
However, 2x = cos θ ⇒
Hence, u = sin–1(sin 2θ) = π – 2θ.
⇒ u = π – 2cos–1(2x)
On differentiating u with respect to x, we get
We know and derivative of a constant is 0.
However,
Now, we have
On differentiating v with respect to x, we get
We know
We know and derivative of a constant is 0.
We have
Thus,