4 : Kvotientregelen

\begin{align}\Big(\frac{x^2}{e^x}\Big)'&=\frac{2x\cdot e^x-x^2 \cdot e^x}{(e^x)^2}\\&=\frac{e^x(2x-x^2) }{(e^x)^2}\\ &=\frac{x(2+x)}{e^x}\end{align}

\begin{align}\bigg(\frac{\ln x}{x}\bigg)'&=\frac{\frac{1}{x}\cdot x-\ln x\cdot 1}{x^2}\\ &=\frac{1-\ln x}{x^2}\end{align}

\begin{align}\bigg(\frac{x}{e^x}\bigg)'&=\frac{1\cdot e^x-x\cdot e^x}{(e^x)²}\\&=\frac{e^x(1-x)}{(e^x)²}\\&=\frac{1-x}{e^x}\end{align}