\[ \mathrm{e}^x = \exp(x) = \ln^{-1}(x) \qquad \int\limits \mathrm{e}^x\,\mathrm{d}x = \mathrm{e}^x + C \qquad \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^x = \mathrm{e}^x \]
- Calculate the derivatives of the following functions explicitly: \[ \newcommand{\ex}{{\mathrm{e}}} f(x) = \ex^{5x^2+1} \qquad g(x) = 3\left(\ex^x+1\right)^2 \qquad h(x) = {\tan(\ex^x)} \]
- Calculate the following integrals explicitly: \[\newcommand{\ex}{\mathrm{e}} \int \left(\ex^{5x} + \pi\right) \,\mathrm{d}x \qquad \int 10x^4\ex^{x^5} \,\mathrm{d}x \qquad \int \ex^x\cos(\ex^x) \,\mathrm{d}x \]