Question

Let $y=\left|\tan \left(\frac{\pi }{4}-x\right)\right|$. Then $\frac{dy}{dx}$ at $x=\frac{\pi }{4}$

• A. is $1$
• B. is $-1$
• C. does not exist
• D. none of these

Answer 1

Answer: (C)

does not exist

$y=\left|\tan (\frac{\pi }{4}-x)\right|\Rightarrow y=\{\begin{array}{c}\begin{array}{cc}\tan (\frac{\pi }{4}-x)& x\ge 0\end{array}\\ \begin{array}{cc}-\tan (\frac{\pi }{4}-x)& x<0\end{array}\end{array}$

$\frac{dy}{dx}=\{\begin{array}{c}\begin{array}{cc}-{\sec }^2(\frac{\pi }{4}-x)& x\ge 0\end{array}\\ \begin{array}{cc}{\sec }^2(\frac{\pi }{4}-x)& x<0\end{array}\end{array}$

For $x=\frac{\pi }{4}$

$\frac{dy}{dx}=\{\begin{array}{c}\begin{array}{cc}-1& x\ge 0\end{array}\\ \begin{array}{cc}1& x<0\end{array}\end{array}$

Hence $\frac{dy}{dx}$ does not exits