Q118 / 170 · Manipal (MET) 2018 · Mathematics

If $f(x)=\sec x-\cot x$, then $f^{\prime}\left(\frac{\pi}{6}\right)$ is equal to

  • A
    -3
  • Correct answer
    B
    14/3
  • C
    11/3
  • D
    5

Solution

Given, $f(x)=\sec x-\cot x$ $\therefore \quad f^{\prime}(x)=\sec x \tan x+\operatorname{cosec}^2 x$ $\begin{aligned} \Rightarrow \quad f^{\prime}\left(\frac{\pi}{6}\right) & =\sec \frac{\pi}{6} \tan \frac{\pi}{6}+\operatorname{cosec}^2 \frac{\pi}{6} \\ & =\frac{2}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}}+4=\frac{14}{3}\end{aligned}$

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

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