Question

If \(f(x)\) and \(g(x)\) are inverse functions of each other and \(f(x) = x+e^x\), then \(g^'(x) =\)...

• A. \(\frac{1}{1+e^{g(x)}}\)
• B. \(\frac{e^{g(x)}}{1+e^{g(x)}}\)
• C. \(\frac{1}{1+g(x)}\)
• D. \(e^{g(x)}\)

Hint:

The derivative of an inverse function is the reciprocal of the derivative of the original at the matching point.

Answer 1

The Correct Option is A

Step 1: Understand the concept
If \(g\) is the inverse of \(f\), then \(f(g(x)) = x\). Differentiating gives \(f'(g(x))\cdot g'(x) = 1\), so \(g'(x) = \dfrac{1}{f'(g(x))}\).

Step 2: Find f'
\(f(x) = x + e^x\), so \(f'(x) = 1 + e^x\).

Step 3: Substitute
Evaluate \(f'\) at \(g(x)\): \(f'(g(x)) = 1 + e^{g(x)}\).

Step 4: Result
\[ g'(x) = \frac{1}{1 + e^{g(x)}} \]
Option (B) has an extra \(e^{g(x)}\) in the numerator, option (C) uses \(g(x)\) instead of \(e^{g(x)}\), and option (D) drops the denominator. Only (A) matches.

Final Answer:
g'(x) equals 1/(1 + e^{g(x)}). This is option (A). \[ \boxed{\text{(A) }\frac{1}{1+e^{g(x)}}} \]