Question

"If e^x+e^y=e^x+y,prove that dy/dx=-e^y-x."

Answer 1

\[e^x+e^y=e^{x+y}\]
By implicit differentiation, you get
\[e^x+e^y\frac{dy}{dx}=e^{x+y}\left(1+\frac{dy}{dx}\right)\]
Solve for \(\dfrac{dy}{dx}\).
In order to prove what you have to prove, you also have to make use of the given info: that \(e^x+e^y=e^{x+y}\).