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JEE Main & Advanced Mathematics Differentiation Question Bank Derivative at a point Standard Differentiation

  • question_answer
    If \[y={{\cos }^{-1}}\cos (|x|-f(x)),\] where            \[\]   \[f(x)\left\{ \begin{align}   & =1\,,\,\text{if}\,\,\,x>0 \\  & =-1\,,\,\text{if}\,\,\,x<0 \\  & =0\,,\,\text{if}\,\,\,x=0 \\ \end{align} \right.\],  then \[{{\left. \frac{dy}{dx} \right|}_{x=\frac{5\pi }{4}}}\]is    [J & K 2005]

    A)            ? 1

    B)            1

    C)            0

    D)            Indeterminate

    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 \[\,\,\frac{5\pi }{4}\in \,[\,1,\,\,\pi +1\,]\]                    \[{{\left. \frac{dy}{dx} \right|}_{x=\frac{5\pi }{4}}}={{\left. \begin{align}   & 1 \\  &  \\ \end{align} \right|}_{x=\frac{5\pi }{4}}}=1\].


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