Question

# Find derivatives for the functions. Assume a, b, c, and k are constants. y=t^{2}e^{5t}

Derivatives
Find derivatives for the functions. Assume a, b, c, and k are constants.
$$\displaystyle{y}={t}^{{{2}}}{e}^{{{5}{t}}}$$

2021-05-05
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