Question

The derivative of `sin^-1(2xsqrt(1 - x^2))` with respect to `sin^-1(3x - 4x^3)` is ______ 

Options

• `2/3`

• `3/2`

• `1/2`

• 1

Answer 1

The derivative of `sin^-1(2xsqrt(1 - x^2))` with respect to `sin^-1(3x - 4x^3)` is `underline(2/3)`.

Explanation:

Let `y = sin^-1(2xsqrt(1 - x^2))`

and z = `sin^-1(3x - 4x^3)`

Put x = sin θ ⇒ θ = `sin^-1x`

∴ `y = sin^-1(2sinthetasqrt(1 - sin^2theta))` and

z = `sin^-1(3sin theta - 4sin^3theta)`

⇒ `y = sin^-1(sin2theta)` and z = `sin^-1(sin3theta)`

⇒ `y = 2theta = 2sin^-1x` and z = `3theta = 3sin^-1x`

∴ `dy/dx = 2/sqrt(1 - x^2)` and `dz/dx = 3/sqrt(1 - x^2)`

∴ `dy/dz = (dy/dx)/(dz/dx) = 2/3`