Find derivative of given function:

\(\displaystyle{y}'=^{{{1}}}{\left({t}^{{{2}}}\right)}'\dot{{e}}^{{{5}{t}}}+{t}^{{{2}}}\dot{{{e}^{{{5}{t}}}}}'\)

\(\displaystyle=^{{{2}}}{2}{t}\dot{{e}}^{{{5}{t}}}+{t}^{{{2}}}\dot{{e}}^{{{5}{t}}}\dot{{{5}{t}}}'\)

\(\displaystyle={2}{t}{e}^{{{5}{t}}}+{5}{t}^{{{2}}}{e}^{{{5}{t}}}\)

\(\displaystyle={\left({2}{t}+{5}{t}^{{{2}}}\right)}{e}^{{{5}{t}}}\)

Rules used:

(1) Product rule

(2) Chain rule

\(\displaystyle{y}'=^{{{1}}}{\left({t}^{{{2}}}\right)}'\dot{{e}}^{{{5}{t}}}+{t}^{{{2}}}\dot{{{e}^{{{5}{t}}}}}'\)

\(\displaystyle=^{{{2}}}{2}{t}\dot{{e}}^{{{5}{t}}}+{t}^{{{2}}}\dot{{e}}^{{{5}{t}}}\dot{{{5}{t}}}'\)

\(\displaystyle={2}{t}{e}^{{{5}{t}}}+{5}{t}^{{{2}}}{e}^{{{5}{t}}}\)

\(\displaystyle={\left({2}{t}+{5}{t}^{{{2}}}\right)}{e}^{{{5}{t}}}\)

Rules used:

(1) Product rule

(2) Chain rule