Explanation Provide the Chain Rule for functions of one independent variable and two intermediate variables. If w = f ( x , y ) is differentiable and if x = x ( t ) , y = y ( t ) are differentiable functions of t , then the composition w = f ( x ( t ) , y ( t ) ) is differentiable function of t and d w d t = f x ( x ( t ) , y ( t ) ) x ′ ( t ) + f y ( x ( t ) , y ( t ) ) y ′ ( t ) d w d t = ∂ f ∂ x d x d t + ∂ f ∂ y d y d t Sketch the diagram to find d w d t for functions of one independent variable and two intermediate variables as shown in Figure 1. Provide the Chain Rule for functions of one independent variable and three intermediate variables. If w = f ( x , y , z ) is differentiable and x , y , and z are differentiable functions of t , then w is a differentiable function of t and d w d t = ∂ f ∂ x d x d t + ∂ f ∂ y d y d t + ∂ f ∂ z d z d t Sketch the diagram to find d w d t for functions of one independent variable and three intermediate variables as shown in Figure 2. The dependency diagram provides a convenient way to remember the Chain Rule. The “true” independent variable in the composite function is t , whereas x and y are intermediate variables and w is the dependent variable. Sketch the composite function and dependency diagrams as shown in Figure 3

Thomas' Calculus: Early Transcendentals, Single Variable (14th Edition)

14th Edition

ISBN: 9780134439419

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 14, Problem 13GYR

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Provide notes on general Chain Rule, the form does it take for functions of two independent variables and three independent variables, the functions defined on surfaces, the diagram of these different forms with example, and the pattern enables one to remember all the different forms.

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Chapter 14, Problem 12GYR

Chapter 14, Problem 14GYR

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Thomas' Calculus: Early Transcendentals, Single Variable (14th Edition)

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