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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Differentiation

NDA 2018 · Mathematics · Differentiation

What is the derivative of the function f(x)=e^ x + ( x)-e^ x at...

What is the derivative of the function f(x)=etan⁡x+ln⁡(sec⁡x)−eln⁡x at x=π4?\mathrm{f}(\mathrm{x})=\mathrm{e}^{\tan \mathrm{x}}+\ln (\sec \mathrm{x})-\mathrm{e}^{\ln \mathrm{x}} \text{ at } \mathrm{x}=\frac{\pi}{4} ?

Options

  1. Ae2\frac{\mathrm{e}}{2}
  2. Be
  3. C2e
  4. D4e

Correct answer

C. 2e

Step-by-step solution

f(x)=etan⁡x+ln⁡(sec⁡x)−eln⁡xf′(x)=etan⁡x⋅sec⁡2x+1sec⁡x⋅sec⁡xtan⁡x−1(∵eln⁡x=x and  ddx(x)=1)f′(π4)=etan⁡π4⋅sec⁡2π4+tan⁡π4−1=2e+1−1=2e\begin{aligned} & f(x)=e^{\tan x}+\ln (\sec x)-e^{\ln x} \\ & f^{\prime}(x)=e^{\tan x} \cdot \sec ^2 x+\frac{1}{\sec x} \cdot \sec x \tan x-1 \\ & \qquad\left(\because e^{\ln x}=x \text{\text{ and } } \frac{d}{d x}(x)=1\right) \\ & f^{\prime}\left(\frac{\pi}{4}\right)=e^{\tan \frac{\pi}{4}} \cdot \sec ^2 \frac{\pi}{4}+\tan \frac{\pi}{4}-1=2 \mathrm{e}+1-1 \\ & =2 \mathrm{e}\end{aligned}
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