Q. 185.0( 1 Vote )
Differentiate 
with respect to 
if 0 < x < 1.
Answer :
Let 
and 
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(sinθ)
Given, 0 < x < 1 ⇒ x ϵ (0, 1)
However, x = cos θ
⇒ cos θ ϵ (0, 1)
Hence, u = sin–1(sinθ) = θ
⇒ u = cos–1x
On differentiating u with respect to x, we get
We know
Now, we have 
By substituting x = cos θ, we have
[∵ sin2θ + cos2θ = 1]
⇒ v = cot–1(cotθ)
However, 
Hence, v = cot–1(cotθ) = θ
⇒ v = cos–1x
On differentiating v with respect to x, we get
We know
We have 
Thus, 
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