What is the d/(dt)(t^ne^{at}) ?

The d/(dt)(t^ne^{at}) is nt^{n-1}e^{at}+e^{at}at^n

What is the first d/(dt)(t^ne^{at}) ?

The first d/(dt)(t^ne^{at}) is nt^{n-1}e^{at}+e^{at}at^n

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\frac{d}{d t} (t^{n} e^{a t})

n t^{n - 1} e^{a t} + e^{a t} a t^{n}

Solution steps

\frac{d}{d t} (t^{n} e^{a t})

Apply the Product Rule:

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

f = t^{n}, g = e^{a t}

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

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

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

= n t^{n - 1} e^{a t} + e^{a t} a t^{n}

\int (3 x + 4)^{2} d x

x \to 2 lim (\frac{∣ x - 2 ∣}{2 - x})

x \to 0 lim (- 2 x^{2})

\frac{2}{s ( s + 2 ) ( s + 3 )}

n = 1 \sum \infty (- \frac{2}{3})^{n - 1}

What is the d/(dt)(t^ne^{at}) ?
The d/(dt)(t^ne^{at}) is nt^{n-1}e^{at}+e^{at}at^n
What is the first d/(dt)(t^ne^{at}) ?
The first d/(dt)(t^ne^{at}) is nt^{n-1}e^{at}+e^{at}at^n