What is the d/(dt)(Ate^{-t/3}) ?

The d/(dt)(Ate^{-t/3}) is A(e^{-t/3}-(e^{-t/3}t)/3)

What is the first d/(dt)(Ate^{-t/3}) ?

The first d/(dt)(Ate^{-t/3}) is A(e^{-t/3}-(e^{-t/3}t)/3)

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

A (e^{- \frac{t}{3}} - \frac{e ^{- \frac{t}{3}} t}{3})

Solution steps

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

Take the constant out:

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

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

Apply the Product Rule:

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

f = t, g = e^{- \frac{t}{3}}

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

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

\frac{d}{d t} (e^{- \frac{t}{3}}) = - \frac{1}{3} e^{- \frac{t}{3}}

= A (1 \cdot e^{- \frac{t}{3}} + (- \frac{1}{3} e^{- \frac{t}{3}}) t)

Simplify

A (1 \cdot e^{- \frac{t}{3}} + (- \frac{1}{3} e^{- \frac{t}{3}}) t) : A (e^{- \frac{t}{3}} - \frac{e ^{- \frac{t}{3}} t}{3})

= A (e^{- \frac{t}{3}} - \frac{e ^{- \frac{t}{3}} t}{3})

sin (x)

(1, - 2), (4, 0)

x \to 6 + lim (\frac{1}{x - 6})

\int \frac{1}{( x + 4 ) ^{2} ( x - 2 )} d x

tan (x)

What is the d/(dt)(Ate^{-t/3}) ?
The d/(dt)(Ate^{-t/3}) is A(e^{-t/3}-(e^{-t/3}t)/3)
What is the first d/(dt)(Ate^{-t/3}) ?
The first d/(dt)(Ate^{-t/3}) is A(e^{-t/3}-(e^{-t/3}t)/3)