Explanation Given information : f g ( x ) = f ( x ) g ( x ) is a function where g ( x ) ≠ 0 . Formula used : f ′ ( x ) = lim Δ x → 0 f ( x + Δ x ) − f ( x ) Δ x Calculation : f ′ ( x ) = lim Δ x → 0 f ( x + Δ x ) − f ( x ) Δ x Replace f by f g in above equation, ( f g ) ′ ( x ) = lim Δ x → 0 f g ( x + Δ x ) − f g ( x ) Δ x Simplifying right-hand expression, f g ( x + Δ x ) − f g ( x ) Δ x = f ( x + Δ x ) g ( x + Δ x ) − f ( x ) g ( x ) Δ x = f ( x + Δ x ) g ( x ) − g ( x + Δ x ) f ( x ) Δ x g ( x + Δ x ) g ( x ) = f ( x + Δ x ) g ( x ) − g ( x + Δ x ) f ( x ) Δ x Δ x g ( x + Δ x ) g ( x ) Δ x [ Divide by <

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 3.2, Problem 56PS

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To prove : (fg)′=gf′−fg′g2

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Chapter 3.2, Problem 55PS

Chapter 3.2, Problem 57PS

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u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (ū+v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅w) Support your answer mathematically or a with a written explanation. d) If possible, find u. (vxw) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

Chapter 3 Solutions

Calculus

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To prove : ( f g ) ′ = g f ′ − f g ′ g 2

Solution Summary: The author explains how to prove that (fg)prime is a function.

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To prove : (fg)′=gf′−fg′g2

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Chapter 3 Solutions