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

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

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

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

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

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

Solution steps

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

Take the constant out:

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

= A \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}

= A (\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}

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

Simplify

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

y^{^{'}} + 7 (\frac{y}{x}) = x

\frac{d}{d x} (\frac{10}{1 + e ^{- 3 x}})

\frac{d}{d x} (e^{2 x} \cdot 4)

y = x, y = \frac{1}{2} x, x = 25

sin (6 (t - \frac{π}{2}))

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