Homework Help is Here – Start Your Trial Now!
Math
Calculus
Let r(x) = f(g(h(x))), where h(1) = 5, g(5) = 2, h'(1) = 9, g'(5) = -2, and f'(2) = 5. Find r'(1).

Let r(x) = f(g(h(x))), where h(1) = 5, g(5) = 2, h'(1) = 9, g'(5) = -2, and f'(2) = 5. Find r'(1).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question
**Problem Statement:** Let \( r(x) = f(g(h(x))) \), where \( h(1) = 5 \), \( g(5) = 2 \), \( h'(1) = 9 \), \( g'(5) = -2 \), and \( f'(2) = 5 \). Find \( r'(1) \). **Solution:** Using the chain rule for differentiation, we have: \[ r(x) = f(g(h(x))) \] Let: - \( u = h(x) \) - \( v = g(u) \) Then: \[ r(x) = f(v) \] By the chain rule: \[ r'(x) = f'(v) \cdot g'(u) \cdot h'(x) \] Evaluate at \( x = 1 \): - \( u = h(1) = 5 \) - \( v = g(u) = g(5) = 2 \) Therefore: - \( h'(1) = 9 \) - \( g'(5) = -2 \) - \( f'(2) = 5 \) Now, substituting these values: \[ r'(1) = f'(2) \cdot g'(5) \cdot h'(1) \] \[ r'(1) = 5 \cdot (-2) \cdot 9 \] \[ r'(1) = 5 \cdot (-18) \] \[ r'(1) = -90 \] Therefore, the value of \( r'(1) \) is: \[ \boxed{-90} \]
Transcribed Image Text:**Problem Statement:** Let \( r(x) = f(g(h(x))) \), where \( h(1) = 5 \), \( g(5) = 2 \), \( h'(1) = 9 \), \( g'(5) = -2 \), and \( f'(2) = 5 \). Find \( r'(1) \). **Solution:** Using the chain rule for differentiation, we have: \[ r(x) = f(g(h(x))) \] Let: - \( u = h(x) \) - \( v = g(u) \) Then: \[ r(x) = f(v) \] By the chain rule: \[ r'(x) = f'(v) \cdot g'(u) \cdot h'(x) \] Evaluate at \( x = 1 \): - \( u = h(1) = 5 \) - \( v = g(u) = g(5) = 2 \) Therefore: - \( h'(1) = 9 \) - \( g'(5) = -2 \) - \( f'(2) = 5 \) Now, substituting these values: \[ r'(1) = f'(2) \cdot g'(5) \cdot h'(1) \] \[ r'(1) = 5 \cdot (-2) \cdot 9 \] \[ r'(1) = 5 \cdot (-18) \] \[ r'(1) = -90 \] Therefore, the value of \( r'(1) \) is: \[ \boxed{-90} \]
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS