Differentiation Application

Thanks to those who responded yesterday to my question regarding the differentiation in this problem. As it turns out, I wasn't off as far as I thought I might be. I believed that correctly differentiating the function was the root of my problem, but as it turns out, I'm not sure how to apply the derivative correctly. Here is a restatement of the problem fleshed out:

The expression for the charge entereing the upper terminal of a basic,

2-terminal circuit element, is given by,

$\displaystyle q=\frac{1}{\alpha^2}-{(\frac{t}{\alpha}+\frac{1}{\alpha^2})}e^{-\alpha t}$ Coulombs.

Find the maximum value of the current entering the terminal if $\displaystyle \alpha=0.03679\ s^{-1}$.

I thought that finding $\displaystyle \lim_{t\rightarrow\infty}\frac{dq}{dt} = te^{-\alpha t}$ would give me the answer. I applied L' Hopital's Rule and ended up with zero, which obviously is incorrect. The book gives an answer of 10 Amperes. If there are any engineers out there, I could use some help. P.S. Amperes = Couloumbs/Second, which is what led me to differentiation in the first place.