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: Understanding the Concept
For the inverse relation, \(\frac{dx}{dy}=\frac{1}{dy/dx}\) and \(\frac{d^{2}x}{dy^{2}}=\frac{d}{dx}\left(\frac{dx}{dy}\right)\cdot\frac{dx}{dy}\).

Step 2: First derivative
\(y=x-e^{x}\Rightarrow\frac{dy}{dx}=1-e^{x}\), so \(\frac{dx}{dy}=\frac{1}{1-e^{x}}\).

Step 3: Differentiate again
\(\frac{d}{dx}\left(\frac{1}{1-e^{x}}\right)=\frac{e^{x}}{(1-e^{x})^{2}}\).

Step 4: Apply the chain rule
\(\frac{d^{2}x}{dy^{2}}=\frac{e^{x}}{(1-e^{x})^{2}}\cdot\frac{1}{1-e^{x}}=\frac{e^{x}}{(1-e^{x})^{3}}\).

Final Answer:
The second derivative is e^x divided by (1 - e^x) cubed. \[ \boxed{\text{(D) }\dfrac{e^{x}}{(1-e^{x})^{3}}} \]