Question

If \(y=x-e^{x}\), then \(\frac{d^{2}x}{dy^{2}}=\) ...

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

Hint:

Find dx/dy as the reciprocal of dy/dx and differentiate again with respect to y using the chain rule.

Answer 1

The Correct Option is D

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

Step 2: Find the derivatives
$\frac{dy}{dx}=1-e^{x}$ and $\frac{d^{2}y}{dx^{2}}=-e^{x}$.

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

Final Answer:
Hence the result is e^x over (1 - e^x) cubed. \[ \boxed{\text{(D) }\dfrac{e^{x}}{(1-e^{x})^{3}}} \]