What is the d/(dt)(t^3e^t) ?

The d/(dt)(t^3e^t) is 3t^2e^t+e^tt^3

What is the first d/(dt)(t^3e^t) ?

The first d/(dt)(t^3e^t) is 3t^2e^t+e^tt^3

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\frac{d}{d t} (t^{3} e^{t})

3 t^{2} e^{t} + e^{t} t^{3}

Solution steps

\frac{d}{d t} (t^{3} e^{t})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = t^{3}, g = e^{t}

= \frac{d}{d t} (t^{3}) e^{t} + \frac{d}{d t} (e^{t}) t^{3}

\frac{d}{d t} (t^{3}) = 3 t^{2}

\frac{d}{d t} (e^{t}) = e^{t}

= 3 t^{2} e^{t} + e^{t} t^{3}