Question

If $e^x+e^y=e^{x+y}$, find $\frac{dy}{dx}$.

Answer 1

Given,

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

Now differentiating both sides with respect to $x$ we get,

$e^x+e^y\frac{dy}{dx}=e^{x+y}(1+\frac{dy}{dx})$

or, $e^x-e^{x+y}=(e^{x+y}-e^y)\frac{dy}{dx}$

or, $\frac{dy}{dx}=e^{x-y}\frac{1-e^y}{e^x-1}$

or, $\frac{dy}{dx}=-e^{x-y}\frac{1-e^y}{1-e^x}$