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