Question

If \(g(x)\) is the inverse of the function \(f(x)\) and \(f'(x) = \dfrac{1}{h(x)}\), then what is the value of \(g'(x)\)?

• A. \(h(g(x))\)
• B. \(g(h(x))\)
• C. \(h'(f(x))\)
• D. \(f(h(x))\)

Hint:

To differentiate an inverse function \(g(x)\), use the identity \(g'(x) = \frac{1}{f'(g(x))}\). Substitute carefully and apply function composition correctly.

Answer 1

The Correct Option is A

We are given that \(g(x)\) is the inverse of \(f(x)\), so by the inverse function rule: \[ g'(x) = \frac{1}{f'(g(x))} \] Given \(f'(x) = \frac{1}{h(x)}\), we substitute \(g(x)\) in place of \(x\): \[ f'(g(x)) = \frac{1}{h(g(x))} \] Therefore, \[ g'(x) = \frac{1}{f'(g(x))} = \frac{1}{\frac{1}{h(g(x))}} = h(g(x)) \]