Question

If $2^x+2^y=2^{x+y}$, then find $\frac{dy}{dx}$

Answer 1

Given $2^x+2^y=2^{x+y}$

$⟹\frac{2^x}{2^{x+y}}+\frac{2^y}{2^{x+y}}=1$

$⟹\frac{1}{2^y}+\frac{1}{2^x}=1$

$⟹2^{-y}+2^{-x}=1$

Differentiating on both sides

$⟹-2^{-y}\frac{dy}{dx}-2^{-x}=0$

$⟹2^{-y}\frac{dy}{dx}=-2^{-x}$

$⟹\frac{dy}{dx}=-2^{y-x}$