$\begin{array}{rl}\text{Produktregel:}\phantom{=}& {(u(x)\cdot v(x))}^{\prime }=u^{\prime }(x)\cdot v(x)+u(x)\cdot v^{\prime }(x)\end{array}$

$\begin{array}{rl}f(x)& =x\cdot \cos (x)\\ f^{\prime }(x)& =1\cdot \cos (x)+x\cdot (-\sin (x))\\ & =\cos (x)-x\cdot \sin (x)\end{array}$

$\begin{array}{rl}\text{Produktregel:}\phantom{=}& {(u(x)\cdot v(x))}^{\prime }=u^{\prime }(x)\cdot v(x)+u(x)\cdot v^{\prime }(x)\end{array}$

$\begin{array}{rl}f(x)& =4x\cdot (x^3+2)\\ f^{\prime }(x)& =4\cdot (x^3+2)+4x\cdot 3x^2\\ & =4x^3+8+12x^3\\ & =16x^3+8\end{array}$

$\begin{array}{rl}\text{Produktregel:}\phantom{=}& {(u(x)\cdot v(x))}^{\prime }=u^{\prime }(x)\cdot v(x)+u(x)\cdot v^{\prime }(x)\end{array}$

$\begin{array}{rl}f(x)& =(\frac{1}{2}{x}^2+2x)\cdot (x-1)\\ f^{\prime }(x)& =(x+2)\cdot (x-1)+(\frac{1}{2}{x}^2+2x)\cdot 1\\ & =x^2-x+2x-2+\frac{1}{2}{x}^2+2x\\ & =\frac{3}{2}{x}^2+3x-2\end{array}$