Question

If $e^y (x+1) = 1$, then find the value of $$ \frac{d^2 y}{dx^2} - \left(\frac{dy}{dx}\right)^2. $$ 

• A. \( e^y \)
• B. \( \frac{1}{x+1} \)
• C. \( -\frac{1}{x+1} \)
• D. \( 0 \)

Hint:

Try simplifying implicit equations before differentiating.

Answer 1

The Correct Option is D

Concept: Use implicit differentiation.
Step 1: Take log. \[ e^y = \frac{1}{x+1} \Rightarrow y = -\ln(x+1) \]
Step 2: First derivative. \[ \frac{dy}{dx} = -\frac{1}{x+1} \]
Step 3: Second derivative. \[ \frac{d^2y}{dx^2} = \frac{1}{(x+1)^2} \]
Step 4: \[ \left(\frac{dy}{dx}\right)^2 = \frac{1}{(x+1)^2} \] \[ \Rightarrow \frac{d^2y}{dx^2} - \left(\frac{dy}{dx}\right)^2 = 0 \]