Question

If \(y = x+e^x\), then \(\frac{d^2x}{dy^2}\) is equal to

• A. \(\frac{e^x}{(1+e^x)^2}\)
• B. \(\frac{-e^x}{(1+e^x)^2}\)
• C. \(\frac{e^x}{(1+e^x)^3}\)
• D. \(\frac{-e^x}{(1+e^x)^3}\)

Hint:

First find dx/dy as the reciprocal of dy/dx, then differentiate again with respect to y using the chain rule.

Answer 1

The Correct Option is D

Step 1: Use the formula for inverse functions.
$\dfrac{d^2x}{dy^2} = -\dfrac{d^2y/dx^2}{(dy/dx)^3}$.

Step 2: Compute the derivatives.
$\dfrac{dy}{dx} = 1 + e^x$ and $\dfrac{d^2y}{dx^2} = e^x$.

Step 3: Substitute.
\[ \frac{d^2x}{dy^2} = -\frac{e^x}{(1 + e^x)^3} \]

Final Answer:
Option (D). \[ \boxed{\frac{-e^x}{(1+e^x)^3}} \]