Explanation Given: The derivative d d x ∫ x x e t t d t . Concept Used: Second Fundamental Theorem of Calculus stats that if f is continuous on [ a , b ] and we define F ( x ) = ∫ a x f ( t ) d t for all x ∈ [ a , b ] , then F is continuous on [ a , b ] and differentiable on ( a , b ) . Fis antiderivative of f , that is, F ' ( x ) = f ( x ) . If f is continuous on [ a , b ] and u ( x ) is differentiable on [ a , b ] , then for all x ∈ [ a , b ] , d d x ∫ a u ( x ) f ( t ) d t = f ( u ( x ) ) ⋅ u ' ( x ) Calculation: In order to find the derivative d d x ∫ x x e t t d t , use the Second Fundamental Theorem of Calculus

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.7, Problem 48S

To determine

The derivative ddx∫xx e tt dt using the Second Fundamental Theorem of Calculus.

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Chapter 4.7, Problem 47S

Chapter 4.7, Problem 49S

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The derivative d d x ∫ x x e t t d t using the Second Fundamental Theorem of Calculus.

Solution Summary: The author explains the second fundamental theorem of calculus, which states that if f is continuous on left[a,bright], then it is differentiable.

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The derivative ddx∫xx e tt dt using the Second Fundamental Theorem of Calculus.

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