Differentiation Question 326
If $ y={{\cos }^{-1}}\cos (|x|-f(x)), $ where $ f(x) $ is a function that ensures the argument of the inverse cosine is within the domain $[-1, 1]$.
$ $ f(x){ \begin{aligned} & =1,\text{ if }x>0 \\ & =-1,\text{ if }x<0 \\ & =0,\text{ if }x=0 \\ \end{aligned} . $ , then $ \left. \frac{dy}{dx} \right| _{x=\frac{5\pi }{4}} $ is
[J & K 2005]
Options:
- 1
1
0
D) Indeterminate
Show Answer
Answer:
Correct Answer: B
Solution:
$ y={{\cos }^{-1}}\cos (x-1),x>0 $
$ \Rightarrow y=x-1, $
$ x>0 $ and $ 0\le x-1\le \pi $
$ \therefore $ $ y=x-1 $ , $ 1\le x\le \pi +1 $
we have, $ 1<\frac{5\pi }{4}<\pi +1 $
$ \therefore y=x-1, $
$ 1\le x\le \pi +1 $ and $ \frac{5\pi }{4}\in [1,\pi +1] $
$ {{. \frac{dy}{dx} |} _{x=\frac{5\pi }{4}}}={{. \begin{vmatrix} & 1 \\ & \\ \end{vmatrix} |} _{x=\frac{5\pi }{4}}}=1 $ .
BETA
