Explanation Given: The function is h ( x , y , z ) = ln ( x + y + z ) , P ( 1 , 0 , 0 ) , Q ( 4 , 3 , 1 ) . Formula used: Gradient is given by, ∇ f ( x , y , z ) = f x ( x , y , z ) i + f y ( x , y , z ) j + f z ( x , y , z ) k Directional derivative is given by, D u f ( x , y , z ) = ∇ f ( x , y , z ) ⋅ u Calculation: Consider the expression h ( x , y , z ) = ln ( x + y + z ) . Partial differentiate h ( x , y , z ) with respect to x , h x ( x , y , z ) = 1 ( x + y + z ) Which is continuous. Partial differentiate h ( x , y , z ) with respect to y , h y ( x , y , z ) = 1 ( x + y + z ) Partial differentiate h ( x , y , z ) with respect to z , h z ( x , y , z ) = 1 ( x + y + z ) Which is continuous. Gradient is given by, ∇ f ( x , y , z ) = f x ( x , y , z ) i + f y ( x , y , z ) j + f z ( x , y , z ) k Now put, x = 1 y = 0 z = 0

Calculus: Early Transcendental Functions

7th Edition

ISBN: 9781337552516

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Question

Chapter 13.6, Problem 28E

To determine

To calculate: The directional derivative of the given function at P in the direction of PQ⇀.

where the function is h(x,y,z)=ln(x+y+z), P(1,0,0), Q(4,3,1).

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Chapter 13.6, Problem 27E

Chapter 13.6, Problem 29E

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Solve part (b) only

Chapter 13 Solutions

Calculus: Early Transcendental Functions

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The directional derivative of the given function at P in the direction of P Q ⇀ . where the function is h ( x , y , z ) = ln ( x + y + z ) , P ( 1 , 0 , 0 ) , Q ( 4 , 3 , 1 ) .

Solution Summary: The author calculates the directional derivative of the given function at P in the direction of stackrelrightharpoonup

Concept explainers

To calculate: The directional derivative of the given function at P in the direction of PQ⇀.

where the function is h(x,y,z)=ln(x+y+z), P(1,0,0), Q(4,3,1).

Want to see the full answer?

Chapter 13 Solutions

The directional derivative of the given function at P in the direction of P Q ⇀ . where the function is h ( x , y , z ) = ln ( x + y + z ) , P ( 1 , 0 , 0 ) , Q ( 4 , 3 , 1 ) .

Solution Summary: The author calculates the directional derivative of the given function at P in the direction of stackrelrightharpoonup

Concept explainers

To calculate: The directional derivative of the given function at P in the direction of PQ⇀. where the function is h(x,y,z)=ln(x+y+z), P(1,0,0), Q(4,3,1).

Want to see the full answer?

Chapter 13 Solutions

To calculate: The directional derivative of the given function at P in the direction of PQ⇀.

where the function is h(x,y,z)=ln(x+y+z), P(1,0,0), Q(4,3,1).