In x'(t) = 3t^2 + 2

x'(t) means the first derivative of x with resepct to t.
That is, the rate at which x changes as t changes.

x''(t) would be the second derivative.
x'''(t) would be the third.

The caret symbolizes exponentiation which is difficult to represent on a computer otherwise. In mathematica, it might be possible to do it. I'm not sure. Old versions of basic used "**", so one would write x'(t) = 3*t**2 + 2 where one asterisk denoted multiplication and two exponentiation.

As that page was not written for a program, but for a human to read, he left out the single asterisk, because most people would understand that better.

Smoothing in relation to functions means trying to come up with parameters that make a function more "natural" looking. There's an art to it, as well as a science. We had to do splines in graduate numerical analysis, which is trying to find a smooth curve to fit n points. I haven't used it since, but I'm vaguely aware that they use splines in some kinds of graphics applications.

Smoothing over age and time should be a property of a function that gets smoother as t increases.