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

  • question_answer
    If \[y=x\text{ }\left[ \left( \cos \frac{x}{2}+\sin \frac{x}{2} \right)\text{ }\left( \cos \frac{x}{2}-\sin \frac{x}{2} \right)+\sin x \right]+\frac{1}{2\sqrt{x}}\], then \[\frac{dy}{dx}=\]

    A)            \[(1+x)\cos x+(1-x)\sin x-\frac{1}{4x\sqrt{x}}\]

    B)            \[(1-x)\cos x+(1+x)\sin x+\frac{1}{4x\sqrt{x}}\]

    C)            \[(1+x)\cos x+(1+x)\sin x-\frac{1}{4x\sqrt{x}}\]

    D)            None of these

    Correct Answer: A

    Solution :

               \[y=x\,\left[ \left( \cos \frac{x}{2}+\sin \frac{x}{2} \right)\,\left( \cos \frac{x}{2}-\sin \frac{x}{2} \right)+\sin x \right]\]\[+\frac{1}{2\sqrt{x}}\]            Þ\[y=x(\cos x+\sin x)+\frac{1}{2\sqrt{x}}\]            Differentiating w.r.t. x, we have            \[\frac{dy}{dx}=x\frac{d}{dx}(\cos x+\sin x)+(\cos x+\sin x)-\frac{1}{4}{{x}^{-3/2}}\]            Þ\[\frac{dy}{dx}=\frac{x\left[ -\frac{1}{2\sqrt{{{a}^{2}}-{{x}^{2}}}}(-2x) \right]-(a-\sqrt{{{a}^{2}}-{{x}^{2}})}}{{{x}^{2}}}\].


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