Question

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

• 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} $
$\Rightarrow \frac{ dy }{ dx }=\frac{2^{x+y}-2^{x}}{2^{y}-2^{x+y}}$
$\Rightarrow \frac{ dy }{ dx }=\frac{2^{x}+2^{y}-2^{x}}{2^{y}-2^{x}-2^{y}}=-2^{y-x}$