\(\displaystyle{H}'{\left({t}\right)}={\left[{\left({a}{t}^{{{2}}}+{b}\right)}{e}^{{-{c}{t}}}\right]}'\)

\(\displaystyle={\left[{\left({a}{t}^{{{2}}}+{b}\right)}\right]}'{e}^{{-{c}{t}}}+{\left({a}{t}^{{{2}}}+{b}\right)}{\left[{e}^{{-{c}{t}}}\right]}'\)

\(\displaystyle={2}{a}{t}{e}^{{-{c}{t}}}-{c}{\left({a}{t}^{{{2}}}+{b}\right)}{e}^{{-{c}{t}}}\)

\(\displaystyle={\left[{\left({a}{t}^{{{2}}}+{b}\right)}\right]}'{e}^{{-{c}{t}}}+{\left({a}{t}^{{{2}}}+{b}\right)}{\left[{e}^{{-{c}{t}}}\right]}'\)

\(\displaystyle={2}{a}{t}{e}^{{-{c}{t}}}-{c}{\left({a}{t}^{{{2}}}+{b}\right)}{e}^{{-{c}{t}}}\)