Continuity and Differentiability — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability2 Marks
Question
Find the second-order derivative of the function x.cos x
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Answer
Let y = x.cos x $\therefore \frac{{dy}}{{dx}} = x\frac{d}{{dx}}\cos x + \cos x\frac{d}{{dx}}x = - x\sin x + \cos x$ $\Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = \frac{d}{{dx}}\left( {\frac{{dy}}{{dx}}} \right) = - \frac{d}{{dx}}\left( {x\sin x} \right) + \frac{d}{{dx}}\cos x$ $= - \left[ {x\frac{d}{{dx}}\sin x + \sin x\frac{d}{{dx}}x} \right] - \sin x$ = -(x cos x + sin x) - sin x = - x cos x - sin x - sin x = -x cos x - 2 sin x = -(x cos x + 2 sin x). Which is the required solution.
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