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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Differentiation

NDA 2019 · Mathematics · Differentiation

Paragraph: Consider the equation x^y=e^ x-y Question: What is (d...

Paragraph: Consider the equation xy=ex−yx^y=e^{x-y}
Question: What is dydx\frac{d y}{d x} at x=1x=1 equal to ?

Options

  1. A0
  2. B1
  3. C2
  4. D4

Correct answer

A. 0

Step-by-step solution

xy=ex−y(dydx)x=1=? at x=1;xy=ex−y⇒1=e1−y⇒y=1∵xy=ex−y⇒y,ln⁡x=x−y\begin{aligned} & x^y=e^{x-y} \\ & \left(\frac{d y}{d x}\right)_{x=1}=? \\ & \text{ at } x=1 ; x^y=e^{x-y} \\ & \Rightarrow 1=e^{1-y} \\ & \Rightarrow y=1 \\ & \because x^y=e^{x-y} \\ & \Rightarrow y, \ln x=x-y \end{aligned} On differentiating w.r.t. xx. ⇒y⋅1x=1−y1−ln⁡x⋅y1⇒1−yx=y1(1+ln⁡x)⇒y1=x−yx(1+ℓnx)∴y1 at x=1=1−11(1+0)=0\begin{aligned} & \Rightarrow y \cdot \frac{1}{x}=1-y^1-\ln x \cdot y^1 \\ & \Rightarrow 1-\frac{y}{x}=y^1(1+\ln x) \\ & \Rightarrow y^1=\frac{x-y}{x(1+\ell \mathrm{n} x)} \\ & \therefore y^1 \text{ at } x=1 \\ & =\frac{1-1}{1(1+0)}=0 \end{aligned}
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