What is the d/(dt)(Ate^{-4t}) ?

The d/(dt)(Ate^{-4t}) is A(e^{-4t}-4e^{-4t}t)

What is the first d/(dt)(Ate^{-4t}) ?

The first d/(dt)(Ate^{-4t}) is A(e^{-4t}-4e^{-4t}t)

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\frac{d}{d t} (A t e^{- 4 t})

A (e^{- 4 t} - 4 e^{- 4 t} t)

Solution steps

\frac{d}{d t} (A t e^{- 4 t})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= A \frac{d}{d t} (t e^{- 4 t})

Apply the Product Rule:

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

f = t, g = e^{- 4 t}

= A (\frac{d t}{d t} e^{- 4 t} + \frac{d}{d t} (e^{- 4 t}) t)

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

\frac{d}{d t} (e^{- 4 t}) = - 4 e^{- 4 t}

= A (1 \cdot e^{- 4 t} + (- 4 e^{- 4 t}) t)

Simplify

= A (e^{- 4 t} - 4 e^{- 4 t} t)

What is the d/(dt)(Ate^{-4t}) ?
The d/(dt)(Ate^{-4t}) is A(e^{-4t}-4e^{-4t}t)
What is the first d/(dt)(Ate^{-4t}) ?
The first d/(dt)(Ate^{-4t}) is A(e^{-4t}-4e^{-4t}t)