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Find the second order derivatives of the functions given in \( x \cos x\)

$\begin{array}{1 1}-x\sin x-\cos x \\-x\cos x-2\sin x \\ -x\sin x+\cos x \\-x\cos x+\cos x \end{array} $

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Toolbox:
  • $\large\frac{d}{dx}$$(x)=1$
  • $\large\frac{d}{dx}$$(\cos x)=\sin x$
Step 1:
Let $y=x\cos x$
$\large\frac{dy}{dx}=$$x\large\frac{d}{dx}$$(\cos x)+\cos x\large\frac{d}{dx}$$(x)$
$\qquad=x(-\sin x)+\cos x.1$
$\qquad=-x\sin x+\cos x$
Step 2:
$\large\frac{d^2y}{dx^2}=-\large\frac{d}{dx}$$(x\sin x)+\large\frac{d}{dx}$$(\cos x)$
$\qquad=-(x\cos x+\sin x.1-\sin x)$
$\qquad=-x\cos x-\sin x-\sin x$
$\qquad=-x\cos x-2\sin x$
answered May 10, 2013 by sreemathi.v
 
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