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*^{ex}. In Leibniz notation,

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.

## Practice:

Find
where *x* > 0. | |

Since *x* is the lower limit of integration, this problem isn't quite as straightforward as some of the others. However, we know that we can flip the limits of integration if we also flip the sign of the expression. This means and so We can move the negative sign outside the derivative operator: Since we now have *x* as the upper limit of integration, we get | |

Find | |

This problem is tricky because instead of having *x* all by itself as the upper limit of integration, we have *x*^{2}: Let
If the problem were asking for just the derivative of *F*(*x*), we would be all set, because we know *F*'(*x*) = sin(*x*^{2}).
Instead, the problem is asking for the derivative of which happens to be the same thing as *F*(*x*^{2}). We can find the derivative of *F*(*x*^{2}) using the chain rule. We have *F* as the outside function and *x*^{2} as the inside function. So Since *F*'(*x*) = sin(*x*^{2}),
we have *F*'(*x*^{2}) = sin((*x*^{2})^{2}) = sin(*x*^{4}).
Putting this back into the earlier equation, | |

Find | |

Now we have the variable *x* showing up in both limits of integration! Thankfully, one of the properties of integrals says we can split the integral up, and the derivative of a sum is the sum of the derivatives, so To deal with the first integral we switch the limits of integration and switch the sign like we did a couple of examples ago: To deal with the other integral we have to use the chain rule, like we did in the previous example. If
then Putting the pieces back together, | |

Find the derivative.

Answer

Find the derivative.

for *t* > 0

Answer

for *t* > 0

Find the derivative.

Answer

Find the derivative.

Answer

Find the derivative.

Answer

Define

Then

Find the derivative.

Answer

First flip the limits of integration and the sign so we have the constant as the lower limit of integration:

If we define

then

Find the derivative.

Answer

In order to take this derivative we first have to split the integral up.

Now we use the chain rule on each piece. If

then

Putting the pieces together,

Since is some constant number, its derivative is 0. So