$\def\arraystretch{1.25} \begin{aligned}\text{Produktregel:}\phantom{=}&\left(\color{blue}{u(x)}\cdot \color{green}{v(x)}\right)'=\color{darkblue}{u'(x)}\cdot \color{green}{v(x)} + \color{blue}{u(x)}\cdot \color{darkgreen}{v'(x)}\end{aligned}$

$\def\arraystretch{1.25} \begin{aligned}f(x) &= \color{blue}{x} \cdot \color{green}{\cos(x)} \\ f'(x) &= \color{darkblue}{1} \cdot \color{green}{\cos(x)} + \color{blue}{x} \cdot \color{darkgreen}{(-\sin(x))} \\ &= \cos(x)-x\cdot\sin(x)\end{aligned}$

$\def\arraystretch{1.25} \begin{aligned}\text{Produktregel:}\phantom{=}&\left(\color{blue}{u(x)}\cdot \color{green}{v(x)}\right)'=\color{blue}{u'(x)}\cdot \color{green}{v(x)} + \color{blue}{u(x)}\cdot \color{green}{v'(x)}\end{aligned}$

$\def\arraystretch{1.25} \begin{aligned}f(x) &= \color{blue}{4x} \cdot \color{green}{(x^3+2)} \\f'(x) &= \color{blue}{4} \cdot \color{green}{(x^3+2)}+\color{blue}{4x} \cdot \color{green}{3x^2} \\&=4x^3+8+12x^3 \\&=16x^3+8\end{aligned}$

$\def\arraystretch{1.25} \begin{aligned}\text{Produktregel:}\phantom{=}&\left(\color{blue}{u(x)}\cdot \color{green}{v(x)}\right)' =\color{blue}{u'(x)}\cdot \color{green}{v(x)} + \color{blue}{u(x)}\cdot \color{green}{v'(x)}\end{aligned}$

$\def\arraystretch{1.25} \begin{aligned}f(x) &= \color{blue}{\left(\frac12x^2+2x\right)} \cdot \color{green}{(x-1)} \\f'(x) &= \color{blue}{(x+2)} \cdot \color{green}{(x-1)}+ \color{blue}{\left(\frac12x^2+2x\right)} \cdot \color{green}{1} \\&=x^2-x+2x-2+\frac12x^2+2x \\&=\frac32x^2+3x-2\end{aligned}$