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: Implicit approach
Set $y = g(x)$, so $x = y + e^y$.

Step 2: Differentiate in x
$1 = \dfrac{dy}{dx}(1 + e^y)$.

Step 3: Solve
$\dfrac{dy}{dx} = \dfrac{1}{1 + e^y} = \dfrac{1}{1 + e^{g(x)}}$.

Step 4: Check
At $x = 1$, $g(1) = 0$ since $f(0) = 1$. Then $g'(1) = \frac{1}{2}$, and $f'(0) = 2$, which agrees with the reciprocal rule.

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