5. Pochodna ilorazu

${\left(\frac{f(x)}{g(x)}\right)}^{\prime }=\frac{f^{\prime }(x)\cdot g(x)-f(x)\cdot g^{\prime }(x{)}^{}}{{\left(g(x)\right)}^2}$

$p\left(x\right)=\frac{2x^2}{\sin x}$

$p^{\prime }\left(x\right)={\left(\frac{2x^2}{\sin x}\right)}^{\prime }=$

$=\frac{(2x^2{)}^{\prime }\cdot \sin x-2x^2\cdot (\sin x{)}^{\prime }}{(\sin x{)}^2}=$

$=\frac{4x\sin x-2x^2\cos x}{{\sin }^{2}x}=$

$=\frac{4x\sin x}{{\sin }^{2}x}-\frac{2x^2\cos x}{{\sin }^{2}x}=$

$=\frac{4x}{\sin x}-\frac{2x^2\mathrm{ctg}x}{\sin x}=\frac{2x}{\sin x}(2-x\mathrm{ctg}x)$