Method of Differentiation
Implicit Differentiation
JEE Main 2013
Grade 12
Question:
If $x^y = e^{x-y}$, then $\frac{dy}{dx}$ is:
(1) $\frac{\ln x}{(1 + \ln x)^2}$
(2) $\frac{1 + \ln x}{(\ln x)^2}$
(3) $\frac{x}{1 + \ln x}$
(4) $\frac{\ln x}{1 + \ln x}$
Step-by-Step Solution
Key Concept: Take logarithms first to convert the relation to $y \ln x = x - y$, then isolate $y = \frac{x}{1 + \ln x}$ before differentiating implicitly.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)