Understanding the fundamental theorem of calculus 
I'm having problems understanding why $$ \frac{d}{dx} \int_{a}^{x} f(t)\,dt = f(x)$$
I'm somewhat weirded out by the fact that there is a $dt$ at the end of $$F(x) = \int_a^x f(t)\,dt$$ too.
We are differentiating with respect to $x$...I understand that $ \frac{d}{dx} \int f(x) \, dx = f(x)$ but why is the $t$ in the definition?

The t means nothing. It is just a dummy variable that allows you to "see" how you are integrating buy it disappears in the end (when we plug in the bounds of integrstion). In the case of F(x), a is some fixed point but x is actually itself variable and is, in fact, the main (and only true) variable in the expression. Thus, we can differentiate with respect to it. The first integral produces F(x)-F(a), and when we differentiate that wrt x, we kill the subtrahend (since a is fixed and thus f(a) is constant too), and get F'(x) which is f(x) by definition. It is a bit tricky the first time.
