If x^y=e^ x-y , then d y / d x at x=1 is - Vidyadip

MCQ

If $x^y=e^{x-y}$, then $d y / d x$ at $x=1$ is

A
$e$
B
$1$
✓
$0$
D
$-1$

✓

Answer

Correct option: C.

$0$

We have, $x^y=e^{x-y}$
Taking logarithm on both sides, we get
$y \log x=x-y$
$ \Rightarrow y(\log x+1)=x$
$\Rightarrow y=\frac{x}{\log x+1} $
$\Rightarrow \frac{d y}{d x}=\frac{1 \cdot(1+\log x)-\frac{1}{x} \cdot x}{(1+\log x)^2}$
$\Rightarrow \frac{d y}{d x}=\frac{1+\log x-1}{(1+\log x)^2}$
$=\frac{\log x}{(1+\log x)^2}$
Now, $\left[\frac{d y}{d x}\right]_{x=1}$
$=\frac{\log 1}{(1+\log 1)^2}=0$

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