Question

If \( e^{x^2y} = c \), then \( \frac{dy}{dx} \) is:

• A. \( \frac{xe^{x^2y}}{2y} \)
• B. \( \frac{-2y}{x} \)
• C. \( \frac{2y}{x} \)
• D. \( \frac{x}{2y} \)

Hint:

For implicit differentiation, always apply logarithmic differentiation when exponentials are involved.

Answer 1

The Correct Option is B

Step 1: Natural Logarithm Application. Given the equation: \[e^{x^2y} = c\] Applying the natural logarithm to both sides yields: \[x^2y = \ln c\] Implicitly differentiating both sides with respect to \( x \): \[2x y + x^2 \frac{dy}{dx} = 0\] Solving for \( \frac{dy}{dx} \): \[\frac{dy}{dx} = \frac{-2y}{x}\] Final Result: The derivative is \( \frac{-2y}{x} \), aligning with option \( \mathbf{(B)} \).