Question

If $e^x+e^y=e^{x+y}$, prove that $\frac{dy}{dx}+e^{y-x}=0$.

Answer 1

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

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

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

$e^x\frac{dy}{dx}+e^y=0$

$\frac{dy}{dx}+e^{y-x}=0$