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a)
$D\left(x^{10}\sin x\right)$
$f\left(x\right)=x^{10}{,}\ f'\left(x\right)=10x^9$
$f\left(x\right)=\sin x{,}\ f'\left(x\right)=\cos x$
$D\left(f\left(x\right)g\left(x\right)\right)=f'\left(x\right)g\left(x\right)+f\left(x\right)g'\left(x\right)$
$D\left(x^{10}\sin x\right)=10x^9\sin x+x^{10}\cos x$
b)
$D\left(x^{10}+\cos x\right)=10x^9-\sin x$
c)
$D\left(\sin x\cos x\right)$
$f\left(x\right)=\sin x{,}\ f'\left(x\right)=\cos x$
$g\left(x\right)=\cos x{,}\ g'\left(x\right)=-\sin x$
$D\left(\sin x\cos x\right)=\cos^2x-\sin^2x$