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. $2^{x-y}.\frac{2^y-1}{1-2^x}$
• D. $\frac{2^{x+y}-2^x}{2^y}$

Answer 1

The correct option is C $2^{x-y}.\frac{2^y-1}{1-2^x}$

Differentiating both sides w.r.t. x,
$2^{x}log2+2^{y}log2\frac{dy}{dx}=2^{x+y}log2(1+\frac{dy}{dx})$
$\therefore 2^x(1-2^y)=2^y(2^x-1)\frac{dy}{dx}$
$\therefore \frac{dy}{dx}=2^{x-y}.\frac{2^y-1}{1-2^x}$