Q56 / 125 · BITSAT 2021 · Mathematics

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

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

Solution

On differentiating $\begin{array}{l} 2^{x} \log 2+2^{y} \log 2 \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \\ =2^{x} \cdot 2^{y} \frac{\mathrm{dy}}{\mathrm{dx}} \cdot \log 2+2^{y} \cdot 2^{x} \log 2 \\ \Rightarrow 2^{x}+2^{y} \frac{\mathrm{dy}}{\mathrm{dx}}=2^{x+y} \frac{\mathrm{dy}}{\mathrm{dx}}+2^{x+y} \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2^{x+y}-2^{x}}{2^{y}-2^{x+y}} \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2^{x}+2^{y}-2^{x}}{2^{y}-2^{x}-2^{y}}=-2^{y-x} \end{array}$

Hence, the correct answer is option: 'D'.

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