Find the second-order derivative of the function x.cos x - Vidyadip

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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