Question

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

• A. \( e^x \)
• 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:

For inverse differentiation, use \( \frac{dx}{dy}=\frac{1}{dy/dx} \) and then apply chain rule carefully for second derivative.

Answer 1

The Correct Option is C

Step 1: Differentiate given equation.
\[ y = x + e^x \]
\[ \frac{dy}{dx} = 1 + e^x \]

Step 2: Find \( \frac{dx}{dy} \).
\[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{1+e^x} \]

Step 3: Differentiate again w.r.t \( y \).
\[ \frac{d^2x}{dy^2} = \frac{d}{dy}\left(\frac{1}{1+e^x}\right) \]

Step 4: Use chain rule.
\[ \frac{d}{dy} = \frac{d}{dx} \cdot \frac{dx}{dy} \]
So:
\[ \frac{d^2x}{dy^2} = \frac{d}{dx}\left(\frac{1}{1+e^x}\right)\cdot \frac{dx}{dy} \]

Step 5: Differentiate w.r.t \( x \).
\[ \frac{d}{dx}\left(\frac{1}{1+e^x}\right) = -\frac{e^x}{(1+e^x)^2} \]

Step 6: Multiply by \( \frac{dx}{dy} \).
\[ \frac{d^2x}{dy^2} = -\frac{e^x}{(1+e^x)^2} \cdot \frac{1}{1+e^x} \]
\[ = -\frac{e^x}{(1+e^x)^3} \]

Step 7: Final conclusion.
\[ \boxed{-\frac{e^x}{(1+e^x)^3}} \]