DERIVATIVE OF AN INTEGRAL

Consider the function F(x) as shown below.

Using the Second Fundamental Theorem of Calculus, the formula to find derivative of F(x) is

F'(x) = (b)'f(b) - (a)'f(a)

Video Lesson

In each case, use the Second Fundamental Theorem of Calculus to finf F'(x).

Problem 1 :

Solution :

Let f(t) = t² - 2t.

F'(x) = (x)'f(x) - (-2)'f(-2)

F'(x) = (1)f(x) - (0)f(-2)

F'(x) = (1)(x² - 2x) - 0

F'(x) = x² - 2x

Problem 2 :

Solution :

F'(x) = (x)'f(x) - (1)'f(1)

F'(x) = (1)f(x) - (0)f(1)

Problem 3 :

Solution :

F'(x) = (x)'f(x) - (1)'f(1)

F'(x) = (1)f(x) - (0)f(1)

Problem 4 :

Solution :

F'(x) = (x)'f(x) - (1)'f(1)

F'(x) = (1)f(x) - (0)f(1)

F'(x) = (1)(√x csc x) - 0

F'(x) = √x csc x

Problem 5 :

Solution :

F'(x) = (4x)'f(4x) - (1)'f(1)

F'(x) = (4)f(4x) - (0)f(1)

F'(x) = 4cot(4x) - 0

F'(x) = 4cot(4x)

Problem 6 :

Solution :

F'(x) = (x²)'f(x²) - (1)'f(1)

F'(x) = (2x)f(x²) - (0)f(1)

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