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

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

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d/(dt)(e^tt)

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

e^{t} t + e^{t}

Solution steps

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

Apply the Product Rule:

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

f = e^{t}, g = t

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

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

\frac{d t}{d t} = 1

= e^{t} t + 1 \cdot e^{t}

Simplify

= e^{t} t + e^{t}