The maclaurin series is:
$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{-6\mathrm{x}},\mathrm{f}\left(0\right)=1\phantom{\rule{0ex}{0ex}}\mathrm{f}\text{'}\left(\mathrm{x}\right)=-6{\mathrm{e}}^{-6\mathrm{x}},\mathrm{f}\text{'}\left(0\right)=-6\left(1\right)=-6\phantom{\rule{0ex}{0ex}}\mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)=36{\mathrm{e}}^{-6\mathrm{x}},\mathrm{f}\text{'}\text{'}\left(0\right)=36\left(1\right)=36\phantom{\rule{0ex}{0ex}}\mathrm{f}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=-216{\mathrm{e}}^{-6\mathrm{x}},\mathrm{f}\text{'}\text{'}\text{'}\left(0\right)=-6\left(1\right)=-216\phantom{\rule{0ex}{0ex}}\mathrm{f}\text{'}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=1296{\mathrm{e}}^{-6\mathrm{x}},\mathrm{f}\text{'}\text{'}\text{'}\text{'}\left(0\right)=1296\left(1\right)=1296$
so,$f\left(x\right)=1-6x+\frac{36}{2!}{x}^{2}-\frac{216}{3!}{x}^{3}+\frac{1296}{4!}{x}^{4}+....$
The maclaurin series of $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{-6\mathrm{x}}$ is:
$f\left(x\right)=1-6x+\frac{36}{2!}{x}^{2}-\frac{216}{3!}{x}^{3}+\frac{1296}{4!}{x}^{4}+....$