Question

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

• A. $2^{x-y}\frac{2^y-1}{2^x-1}$
• B. $2^{x-y}\frac{2^y-1}{1-2^x}$
• C. $\frac{2^x+2^y}{2^x-2^y}$
• D. None of these

Answer 1

Answer: (D)

On differentiating
$2^x\log 2+2^y\log 2\cdot \frac{dy}{dx}$
$=2^x\cdot 2^y\frac{dy}{dx}\cdot \log 2+2^y\cdot 2^x\log 2$
$\Rightarrow 2^x+2^y\frac{dy}{dx}=2^{x+y}\frac{dy}{dx}+2^{x+y}$
$\Rightarrow \frac{dy}{dx}=\frac{2^{x+y}-2^x}{2^y-2^{x+y}}$
$\Rightarrow \frac{dy}{dx}=\frac{2^x+2^y-2^x}{2^y-2^x-2^y}=-2^{y-x}$