Q126 / 200 · Manipal (MET) 2010 · Mathematics

If $x^y=e^{x-y}$, then $\frac{d y}{d x}$ is equal to

  • Correct answer
    A
    $\frac{\log x}{(1+\log x)^2}$
  • B
    $\frac{x-y}{(1+\log x)}$
  • C
    $\frac{x-y}{(1+\log x)^2}$
  • D
    $\frac{1}{(1+\log x)}$

Solution

We have, $x^y=e^{x-y}$ $\Rightarrow y \log x=(x-y) \log e=x-y$ $\Rightarrow y=\frac{x}{1+\log x}$ $\Rightarrow \frac{d y}{d x}=\frac{(1+\log x) \cdot 1-x \cdot \frac{1}{x}}{(1+\log x)^2}=\frac{\log x}{(1+\log x)^2}$

Hence, the correct answer is option: 'A'.

More Differentiation Questions