Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx.
y = (2x3 + 5)4
To find: The composite function in the form
Answer to Problem 2E
The inner function is
The derivative of y is
Explanation of Solution
Given:
The function is
Formula used:
The Chain Rule:
If h is differentiable at x and g is differentiable at
Derivative Rule:
(1) Power Rule:
(2) Sum Rule:
Calculation:
Let the inner function be
Then,
Therefore,
Hence, the inner function is
Thus, the required form of composite function is
Obtain the derivative of y .
Let
Apply the chain rule as shown in equation (1)
The derivative
Apply the power rule (2) then substitute
The derivative
The derivative of
Apply the sum rule (2) and the power rule (1),
Thus, the derivative of
Substitute
Therefore, The derivative of
Chapter 3 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning