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?

http://en.wikipedia.org/wiki/Free_variables_and_bound_variables
The expression
$$
\sum_{i=1}^3 \cos(i^2 k^3)
$$
means
$$
\cos(1^2k^3) + \cos(2^2k^3)+\cos(3^2k^3),
$$
and that's the same as
$$
\sum_{j=1}^3 \cos(j^2 k^3),
$$
i.e. $i$ and $j$ are "bound variables", whereas $k$ is a "free variable".
The $t$ in
$$
\int_a^x f(t)\,dt
$$
is a bound variable, like $i$ and $j$ above, and $x$ is a free variable, like $k$ above.  The value of the sums above depends on $k$, but not on anything called $i$ or $j$, and similarly the value of the integral above depends on $x$, but not on anything called $t$.
