Differentiate with respect to if .
Let u = cos–1(4x3 – 3x) and
We need to differentiate u with respect to v that is find.
We have u = cos–1(4x3 – 3x)
By substituting x = cos θ, we have
u = cos–1(4cos3θ – 3cosθ)
But, cos3θ = 4cos3θ – 3cosθ
⇒ u = cos–1(cos3θ)
Given,
However, x = cos θ
Hence, u = cos–1(cos3θ) = 3θ
⇒ u = 3cos–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 = tan–1(tanθ)
However,
Hence, v = tan–1(tanθ) = θ
⇒ v = cos–1x
On differentiating v with respect to x, we get
We know
We have
Thus,