# 5.3 Integration & Anti-derivatives The Fundamental Theorem of Calculus Question #4 (Medium)

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5.3 Integration & Anti-derivatives

The Fundamental Theorem of Calculus

Question #4 (Medium): Finding the Derivative of the Function

Strategy

Notice that the Fundamental Theorem of Calculus Part 1 has

, where the integral goes

from a numeric value of to a variable . This means that only one endpoint of the interval is a variable.

Two complex variations to this are:

1) One end containing a more complex function instead a simple : Then use Leibniz notation and

substitute a simpler variable. Apply the Chain Rule in conjunction with the Fundamental

Theorem of Calculus Part 1 to find the derivative of the function.

2) Both ends of the integral contain some kind of variable expression: Then split the integral into

two with usually as an easy dividing point, then proceed.

.

Sample Question

Find the derivative of the function.

Solution

Recall Leibniz notation of the Fundamental Theorem of Calculus Part 1

.

Here only one end of the interval contains a variable, so Chain Rule is needed.

Let , then . Rearranging gives

.

Then

Therefore, the derivative of the function is:

is .