If $e^{x+y}=xy,$ what is $\large\frac{dy}{dx}$?

Question

$\begin{array}{1 1}(A)\;\large\frac{y(1-x)}{x(y-1)}\\(B)\;\large\frac{y(1+x)}{x(y-1)}\\(C)\;\large\frac{y(1-x)}{x(1+y)}\\(D)\;\large\frac{y(1+x)}{x(1-y)}\end{array}$

Answer 1

Given $e^{x+y} = xy$

Taking log on both sides, we get: $ (x+y) \log_e e = \log_e x + \log_e y$

$\Rightarrow x+y = \log_e x + \log_e y$

Differentiating, we get: $dx + dy = \large\frac{1}{x}$$dx + \large\frac{1}{y}$$dy$

Rearranging, we get: $\large\frac{dy}{dx}$$(1 - \large\frac{1}{y}$$) = \large\frac{1}{x}$$-1$

$\Rightarrow \large\frac{dy}{dx} $$ = \large\frac{y(1-x)}{x(y-1)}$