Calculus 10th Edition

$f''(x)=2\cos{x}-x\sin{x}$
First Derivative: Product Rule $(f’(x)=(u(x)(v(x))’=u’(x)v(x)+u(x)v’(x))$ $u(x)=x ;u’(x)=1$ $v(x)=\sin{x} ;v’(x)=\cos{x}$ $f'(x)=x\cos{x}+\sin{x}$ Second Derivative: $f'(x)=g(x)+h(x)\rightarrow g(x)=\sin{x}$; $h(x)=x\cos{x}$ Product Rule: $h'(x)=(u(x)(v(x))’=u’(x)v(x)+u(x)v’(x)$ $u(x)=x ;u’(x)=1$ $v(x)=\cos{x} ;v’(x)=-\sin{x}$ $h'(x)=\cos{x}-x\sin{x}.$ $g'(x)=\frac{d}{dx}\sin{x}=\cos{x}.$ $f''(x)=g'(x)+h'(x)=2\cos{x}-x\sin{x}.$