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BITSAT 2021
Exam
BITSAT
Year
2021
Subject
Mathematics
Chapter
Differentiation

BITSAT 2021 · Mathematics · Differentiation

If 2^ x +2^ y =2^ x+y , then (d y)/(d x)=

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

Options

  1. A2x−y2y−12x−12^{x-y} \frac{2^{y}-1}{2^{x}-1}
  2. B2x−y2y−11−2x2^{x-y} \frac{2^{y}-1}{1-2^{x}}
  3. C2x+2y2x−2y\frac{2^{x}+2^{y}}{2^{x}-2^{y}}
  4. DNone of these

Correct answer

D. None of these

Step-by-step solution

On differentiating 2xlog⁡2+2ylog⁡2⋅dydx=2x⋅2ydydx⋅log⁡2+2y⋅2xlog⁡2⇒2x+2ydydx=2x+ydydx+2x+y⇒dydx=2x+y−2x2y−2x+y⇒dydx=2x+2y−2x2y−2x−2y=−2y−x\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}
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