What is the d/(dt)(te^{-t}) ?

The d/(dt)(te^{-t}) is e^{-t}-e^{-t}t

What is the first d/(dt)(te^{-t}) ?

The first d/(dt)(te^{-t}) is e^{-t}-e^{-t}t

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

e^{- t} - e^{- t} t

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

= 1 \cdot e^{- t} + (- e^{- t}) t

Simplify

= e^{- t} - e^{- t} t

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x^{- 8}