Question

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

• A. $\frac{2^x+2^y}{2^x-2^y}$
• B. $\frac{2^x+2^y}{1+2^{x+y}}$
• C. $2^{x-y}\left(\frac{2^y-1}{1-2^x}\right)$
• D. $\frac{2^{x+y}-2^x}{2^y}$

Answer 1

The correct option is C $2^{x-y}\left(\frac{2^y-1}{1-2^x}\right)$

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

Differentiating both sides

$\ln 2.2^x+\ln 2.2^y\frac{dy}{dx}=\ln 2.2^{x+y}(1+\frac{dy}{dx})2^x+2^y\frac{dy}{dx}=2^{x+y}(1+\frac{dy}{dx})2^x+2^y\frac{dy}{dx}=2^{x+y}+2^{x+y}\frac{dy}{dx}(2^y-2^{x+y})\frac{dy}{dx}=(2^{x+y}-2^x)\frac{dy}{dx}=\frac{2^{x+y}-2^x}{2^y-2^{x+y}}\frac{dy}{dx}=\frac{2^x(2^y-1)}{2^y(1-2^x)}\frac{dy}{dx}=2^{x-y}\left(\frac{2^y-1}{1-2^x}\right)$

So option $C$ is correct.