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Implicit Differentiation

If xy=ex+y ...

Question

If $x^{y} = e^{x + y}$ then $\frac{d y}{d x}$ at $x = 1$ is equal to

A

0

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B

- 2

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C

1

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D

none of these

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Solution

The correct option is D $- 2$
Given, $x^{y} = e^{x + y}$
Taking $log$ on both sides, we get
$y log x = x + y$
Taking derivative,
$\frac{y}{x} + l o g x . y^{'} = 1 + y^{'}$
At, $x = 1 => y = - 1$
So, $y^{'} = - 2$

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