How to solve this?

Question

Differentiate x^sin-1(x) with respect to sin^-1x

Answer 1

Let t = arcsin x

Using ln and e, we can obtain the value of exponential function as:

x^arcsin(x) = x^t = e^t•ln(x)

d(x^t)/dt = d(e^t•ln(x) )/dt

= e^t•ln(x) • (ln(x) + (t / x)•dx/dt )

Put back the value of t in terms of x:

= x^arcsin(x)•( ln(x) + (arcsin(x) / x) • dx / d(arcsin (x))

Using the derivative of arcsin (x): d(arcsin (x)) = dx / (1-x²)^1/2

= x^arcsin(x)• (ln(x) + (arcsin(x) / x) • dx / (dx /(1-x²)^1/2))

= x^arcsin(x)• (ln(x) + (arcsin(x) / x) • (1-x²)^1/2)