Question

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

For inverse functions, always remember: \[ f'(x) = \frac{1}{g'(f(x))} \] where \( g \) is the original function and \( f \) is its inverse.

Answer 1

The Correct Option is A

Step 1: Use the inverse function differentiation formula.
Recall that if \( f \) is the inverse of \( g \), then: \[ f'(x) = \frac{1}{g'\left( f(x) \right)} \] Given: \[ g'(x) = \frac{1}{1+x^n} \] Thus: \[ g'\left( f(x) \right) = \frac{1}{1+\{ f(x) \}^n} \]
Step 2: Substitute into the formula for \( f'(x) \).
Thus: \[ f'(x) = \frac{1}{\frac{1}{1+\{ f(x) \}^n}} = 1+\{ f(x) \}^n \]
Step 3: Conclude the answer.
Hence, \( f'(x) = 1+\{ f(x) \}^n \).