# The Fundamental Theorem of Calculus

### Topics

The second FTC states that a function of the form

is an antiderivative of *f*(*x*). This means *F*'(*x*) = *f*(*x*). Using Leibniz notation,

### Sample Problem

Find

Answer.

The second FTC says that the integral expression

is an antiderivative of *e ^{ex}*. This means the derivative of is

*e*. In Leibniz notation,

^{ex}In the previous example, all we had to do was write down the integrand, replacing *t* with *x*. This works so long as the lower limit of integration is a constant and the upper limit of integration is *x* all by itself.

We could also change the letters around. If we're using *t* as the upper limit of integration, then we have an antiderivative *F*(*t*) and its derivative will also be a function of *t*.

### Sample Problem

Answer.

This question is asking for the derivative of

Since *F* is a function of *t*, its derivative is also a function of *t*. All we have to do is write down the integrand, replacing *x* with *t* so that we end up with a function of *t*.

Using properties of integrals and the chain rule, we can find the derivates of more complicated functions.