Calculus Facts
Derivative of an Integral (Fundamental Theorem of Calculus) 
When a limit of integration is a function of the variable of differentiation The statement of the fundamental theorem of calculus shows the upper limit of the integral as exactly the variable of differentiation. Using the chain rule in combination with the fundamental theorem of calculus we may find derivatives of integrals for which one or the other limit of integration is a function of the variable of differentiation. Example 1: Find
To find this derivative, first write the function defined by the integral as a composition of two functions h(x) and g(x), as follows:
since
The derivative of a composition of two functions is found using the chain rule:
The derivative of h(x) uses the fundamental theorem of calculus, while the derivative of g(x) is easy:
Therefore:
Notice carefully the h'(g(x)) part of the answer: x^{2} replaces x in tan(x^{3}), giving tan((x^{2})^{3}) = tan(x^{6}). We look at another example. Example 2: Find
See if you can provide the answers to the steps leading to the answer. Roll the mouse over each of the following steps once you've decided. The first step is to identify the integral in this example as the composition h(g(x)):
(Remove answer to the above step) Compute the derivatives of h(x) and g(x):
(Remove answer to the above step) Therefore:
(Remove answer to the above step) Once you see the pattern of use of the chain rule you don't have to write out all the intermediate steps. Notice the upper limit replaces the variable of integration wherever it appears in the integrand and the result is multiplied by the derivative of the upper limit:
(This formula literally is just the chain rule, since f is the derivative of its antiderivative (given by the indefinite integral)  in the notation of the earlier examples, h'(x) = f(x).) Example 3:
(Remove answer to the above step) More:

Derivative of an Integral 
Calculus Facts 
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