Finding Derivatives Examples

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 x²:

Let

If the problem were asking for just the derivative of F(x), we would be all set, because we know

F '(x) = sin(x²).

Instead, the problem is asking for the derivative of

which happens to be the same thing as F(x²). We can find the derivative of F(x²) using the chain rule. We have F as the outside function and x² as the inside function. So

Since

F '(x) = sin(x²),

we have

F '(x²) = sin((x²)²) = sin(x⁴).

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,