 # Find derivatives of the functions defined as follows.y=(t^2 e^(2t))/(t+e^(3t)) shadsiei 2020-12-28 Answered

Find derivatives of the functions defined as follows. $y=\left({t}^{2}{e}^{\left(2t\right)}\right)/\left(t+{e}^{\left(3t\right)}\right)$

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$dy/dt=\left(\left(t+{e}^{\left(3t\right)}\right)\left(2{t}^{2}{e}^{\left(2t\right)}+2t{e}^{\left(2t\right)}\right)-\left({t}^{2}{e}^{\left(2t\right)}\right)\left(1+3{e}^{\left(3t\right)}\right)\right)/\left(\left(t+{e}^{\left(3t\right)}{\right)}^{2}\right)$
$=\left(2{t}^{3}{e}^{\left(2t\right)}+2{t}^{2}{e}^{\left(2t\right)}+2{t}^{2}{e}^{\left(5t\right)}+2t{e}^{\left(5t\right)}-{t}^{2}{e}^{\left(2t\right)}+3{t}^{2}{e}^{\left(5t\right)}\right)/\left(\left(t+{e}^{\left(3t\right)}{\right)}^{2}\right)$
$=\left(\left(5{t}^{2}+2t\right){e}^{\left(5t\right)}+\left(2{t}^{3}+{t}^{2}\right){e}^{\left(2t\right)}\right)/\left(\left(t+{e}^{\left(3t\right)}{\right)}^{2}\right)$