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

  • question_answer
    If \[y={{\tan }^{-1}}\left[ \frac{\sin x+\cos x}{\cos x-\sin x} \right]\,,\] then \[\frac{dy}{dx}\] is               [UPSEAT 2001]

    A)            \[1/2\]

    B)            \[\pi /4\]

    C)            0

    D)            1

    Correct Answer: D

    Solution :

               \[y={{\tan }^{-1}}\left[ \frac{\sin x+\cos x}{\cos x-\sin x} \right]\]\[={{\tan }^{-1}}\left[ \frac{1+\tan x}{1-\tan x} \right]\]               \[={{\tan }^{-1}}\left[ \frac{\tan (\pi /4)+\tan x}{1-\tan (\pi /4)\tan x} \right]\]\[={{\tan }^{-1}}\tan (\pi /4+x)\]                    Þ \[y=(\pi /4)+x\]Þ \[dy/dx=1\].


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