I don’t know how to identify what h, t and T is correspondingly in this exercise. Thus, I can’t apply the chain rule. I know this is a basic question but it seems like I can’t wrap my head around this simple exercise, how to break f(x) down to 3 components as shown in the lecture.

First, split up the right side into its two functions: f(x) = e^u, where u = 2x.
By the chain rule, the derivative f'(x) is the derivative of the first function, times the derivative of the 2nd function.
The derivative of e^u is e^u, substitute in for u and you have e^{2x}.
The derivative of 2x is 2.
Multiply the two, and you have 2e^{2x}.