Question

If \( f \) is the inverse function of \( g \) and \( g'(x) = \dfrac{1{1 + x^n} \), then the value of \( f'(x) \) is:}

• A. \( 1 + (f(x))^n \)
• B. \( 1 - (f(x))^n \)
• C. \( \{1 + f(x)\}^n \)
• D. \( (f(x))^n \)

Hint:

Inverse derivative formula: \[ f'(x) = \frac{1}{g'(f(x))} \] Always substitute the inverse inside derivative.

Answer 1

The Correct Option is A

Concept: Derivative of inverse function: \[ f'(x) = \frac{1}{g'(f(x))} \] Step 1: {\color{red}Apply inverse derivative rule.} Given: \[ g'(x) = \frac{1}{1 + x^n} \] So: \[ f'(x) = \frac{1}{g'(f(x))} \] Step 2: {\color{red}Substitute.} \[ g'(f(x)) = \frac{1}{1 + (f(x))^n} \] Hence: \[ f'(x) = 1 + (f(x))^n \]