Q61 / 180 · KCET 2020 · Mathematics

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

  • A
    $2^{y-x}$
  • Correct answer
    B
    $-2^{y-x}$
  • C
    $2^{x-y}$
  • D
    $\frac{2^{y}-1}{2^{x}-1}$

Solution

We have, $2^{x}+2^{y}=2^{x+y}...(i)$ On differentiating Eq. (i) w.r.t. $x$, we get $2^{x} \log 2+2^{y} \log 2 \frac{d y}{d x}$ $=2^{x+y} \log 2\left(1+\frac{d y}{d x}\right)$ $\Rightarrow \quad 2^{x}+2^{y} \frac{d y}{d x}=2^{x+y}\left(1+\frac{d y}{d x}\right)$ $\Rightarrow \quad 2^{x}-2^{x+y}=\frac{d y}{d x}\left(2^{x+y}-2^{y}\right)$ $\Rightarrow \quad-2^{y}=\frac{d y}{d x}\left(2^{x}\right)$ $\Rightarrow \quad \frac{d y}{d x}=-2^{y} / 2^{x}=-2^{y-x}$

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

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