What is the d/(dt)(ate^t) ?

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

What is the first d/(dt)(ate^t) ?

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

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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)

3 x y^{2} y^{^{'}} + 3 y^{3} = 1

\frac{d}{d x} (- 2 e^{- x})

\frac{d}{d x} (1 + x^{2} + x^{6})

y^{^{''}} + y^{^{'}} = 3

f (x) = x^{2} - x