oh yeah... makes sense. We were never really taught this stuff in class, (I don't think they deemed it necessary for IT majors to learn theory.) Thanks a lot!
Actually, I encountered the integral in a contest. The question was:
\frac{d^{2}}{(dx)^{2}}\int^{x}_{0}\left(\int^{sin t}_{1}\sqrt{u^{4}+1} du\right) dt}
I wasn't so sure how to solve it, but what I did was:
(I'm not even sure this is proper use of the fundamental theorem of the...
The first one is found using the rule for differentiating the inverse of a function,
(f^{-1})^{'}(x) = \frac{1}{f^{'}(f^{-1})}
because the exponential function is the inverse of the natural logarithmic function.
so
\frac{d}{dx}e^{x} = \frac{1}{\frac{d}{dx}[ln (e^{x})]}...