\(\displaystyle{v}'{\left({t}\right)}={\left({t}^{{{2}}}{e}^{{-{c}{t}}}\right)}'\)

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

\(\displaystyle={2}{t}{e}^{{-{c}{t}}}-{c}{t}^{{{2}}}{e}^{{-{c}{t}}}\)

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

\(\displaystyle={2}{t}{e}^{{-{c}{t}}}-{c}{t}^{{{2}}}{e}^{{-{c}{t}}}\)