d((1/a^2)-((t/a)+(1/a^2))e^(-at))/dt

asked by guest
on Oct 21, 2024 at 1:53 pm



You asked:

Evaluate the expression: \(\frac{d \cdot \left(\frac{1}{{a}^{2}} - \left(\frac{t}{a} + \frac{1}{{a}^{2}}\right) \cdot {e}^{-\left( a t \right)}\right)}{d t}\)

MathBot Answer:

Evaluated



\(\displaystyle \frac{d \cdot \left(\frac{1}{{a}^{2}} - \left(\frac{t}{a} + \frac{1}{{a}^{2}}\right) \cdot {e}^{-\left( a t \right)}\right)}{d t} = \frac{- \left(\frac{t}{a} + \frac{1}{a^{2}}\right) e^{- a t} + \frac{1}{a^{2}}}{t} \)

Expanded

\[\frac{d \cdot \left(\frac{1}{{a}^{2}} - \left(\frac{t}{a} + \frac{1}{{a}^{2}}\right) \cdot {e}^{-\left( a t \right)}\right)}{d t} = - \frac{e^{- a t}}{a} + \frac{1}{a^{2} t} - \frac{e^{- a t}}{a^{2} t}\]

Factored

\[\frac{d \cdot \left(\frac{1}{{a}^{2}} - \left(\frac{t}{a} + \frac{1}{{a}^{2}}\right) \cdot {e}^{-\left( a t \right)}\right)}{d t} = - \frac{\left(a t - e^{a t} + 1\right) e^{- a t}}{a^{2} t}\]

\(e\) is Euler's number, a mathematical constant that is approximately \(2.71828\).

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