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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

Finding Derivatives

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 eex. This means the derivative of  is eex. 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.

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