Explanation Second Fundamental Theorem of Calculus: If f ( t ) be any continuous function over the interval [ a , b ] and the definite integral is given by G ( x ) = ∫ a x f ( t ) d t for a ≤ t ≤ b Then G is an antiderivative of f ( t ) on [ a , b ] , and the derivative of G ( x ) is given by G ′ ( x ) = d d x [ ∫ a x f ( t ) d t ] = f ( x ) Now, consider the formula. d d x ∫ a u ( x ) f ( t ) d t = f ( u ( x ) ) u ′

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.7, Problem 17C

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Toexplain:the formula ddx∫au(x)f(t)dt=f(u(x))u′(x) as an application of chain rule.

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Chapter 4.7, Problem 16C

Chapter 4.7, Problem 18C

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Toexplain: the formula d d x ∫ a u ( x ) f ( t ) d t = f ( u ( x ) ) u ′ ( x ) as an application of chain rule.

Solution Summary: The author explains how the formula d dxdisplaystyle is an application of chain rule, and the derivative of G is given by

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Toexplain:the formula ddx∫au(x)f(t)dt=f(u(x))u′(x) as an application of chain rule.

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Chapter 4 Solutions