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

  • question_answer
    \[\frac{d}{dx}\left[ \left( \frac{{{\tan }^{2}}2x-{{\tan }^{2}}x}{1-{{\tan }^{2}}2x{{\tan }^{2}}x} \right)\cot 3x \right]\]                 [AMU 2000]

    A)            \[\tan 2x\,\tan x\]

    B)            \[\tan 3x\tan x\]

    C)            \[{{\sec }^{2}}x\]

    D)            \[\sec x\tan x\]

    Correct Answer: C

    Solution :

               Let \[y=\frac{{{\tan }^{2}}2x-{{\tan }^{2}}x}{1-{{\tan }^{2}}2x{{\tan }^{2}}x}\]                     = \[\frac{(\tan 2x-\tan x)}{(1+\tan 2x\tan x)}\,\frac{(\tan 2x+\tan x)}{(1-\tan 2x\tan x)}\]                     = \[\tan (2x-x)\,\tan (2x+x)\] = \[\tan x\tan 3x\].            \  \[\frac{d}{dx}[y.\cot 3x]=\frac{d}{dx}[\tan x]={{\sec }^{2}}x\].


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