Differentiate sin^-1(2ax√(1 - a^2x^2)) with respect to √(1 - a^2 x^2,) if -1/√2 < ax < 1/√2

Question

Differentiate sin^-1\((2a\text x\sqrt{1-a^2\text x^2})\) with respect to \(\sqrt{1-a^2\text x^2},\) if \(-\cfrac{1}{\sqrt2}\) < ax < \(\cfrac{1}{\sqrt2}\).

Answer 1

u = sin^-1\((2a\text x\sqrt{1-a^2\text x^2})\) and v = \(\sqrt{1-a^2\text x^2}.\)

We need to differentiate u with respect to v that is find \(\cfrac{du}{dv}. \)

We have

By substituting ax = sin θ, we have

On differentiating u with respect to x, we get

Now, we have v = \(\sqrt{1-a^2\text x^2}\)

On differentiating v with respect to x, we get

We know \(\cfrac{d}{d\text x} \)(xⁿ) = nx^n-1 and derivative of a constant is 0.

We have