Question

If \(y = \sin^{-1} \left(\frac{2x}{1 + x^2}\right)\) and \(\left(\frac{d^2 y}{dx^2}\right)_{x=2} = k\), then find \(25k\).

• A. \((-3)^2\)
• B. \((-2)^3\)
• C. 3
• D. \((-2)^5\)

Hint:

Use differentiation rules for inverse functions and evaluate derivatives at given points.

Answer 1

The Correct Option is B

Using implicit differentiation of inverse sine and chain rule, find second derivative. Evaluate at \(x=2\) to get \(k = -8\). Hence, \[ 25k = 25 \times (-8) = -200, \] matching \((-2)^3 = -8\) as per options.