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

Step 1: Understanding the Concept:
Since $e^0 = 1$, the equation $e^{y(x+1)} = 1$ implies that the exponent must be zero.
Step 2: Formula Application:
$y(x + 1) = 0 \implies y = 0$ (for $x \neq -1$). Alternatively, taking $\ln$ on both sides: $y(x + 1) = \ln(1) = 0$. Differentiating with respect to $x$: $y + (x + 1)\frac{dy}{dx} = 0$.
Step 3: Explanation:
Since $y = 0$, then $\frac{dy}{dx} = 0$. Differentiating again: $\frac{dy}{dx} + \frac{dy}{dx} + (x + 1)\frac{d^2y}{dx^2} = 0 \implies \frac{d^2y}{dx^2} = 0$. Thus, $0 - (0)^2 = 0$.
Step 4: Final Answer:
The result is 0.