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: Understanding the Concept:
To get \(\dfrac{d^2x}{dy^2}\) we treat \(x\) as a function of \(y\). We first find \(\dfrac{dx}{dy} = \dfrac{1}{dy/dx}\) and then differentiate this again with respect to \(y\).

Step 2: First derivative.
\[ \frac{dy}{dx} = 1 + e^x \Rightarrow \frac{dx}{dy} = \frac{1}{1 + e^x} \]

Step 3: Second derivative.
\[ \frac{d^2x}{dy^2} = \frac{d}{dx}\left(\frac{1}{1 + e^x}\right)\cdot\frac{dx}{dy} \]
\[ \frac{d}{dx}\left(\frac{1}{1 + e^x}\right) = \frac{-e^x}{(1 + e^x)^2} \]
\[ \frac{d^2x}{dy^2} = \frac{-e^x}{(1 + e^x)^2}\cdot\frac{1}{1 + e^x} = \frac{-e^x}{(1 + e^x)^3} \]

Step 4: Why the other options are wrong.
Option (B) forgets the extra factor \(\dfrac{dx}{dy}\), which gives only a square in the denominator. Options (A) and (C) have the wrong sign.

Final Answer:
\(\dfrac{d^2x}{dy^2} = \dfrac{-e^x}{(1+e^x)^3}\), option (D). \[ \boxed{\frac{-e^x}{(1+e^x)^3}} \]