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...
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![**Problem: Calculating Derivative**
**Task:**
Find the derivative with respect to \( x \) of the function:
\[ f(x) = \left( (2x^3 + 4)^5 + 6 \right)^8 + 7. \]
**Solution:**
To find \( f'(x) \), apply the chain rule multiple times due to the composition of functions within the given expression.
1. Start with the outermost function and work your way inward.
2. Identify each layer of functions and differentiate accordingly, applying the chain rule.
**Comments:** The function involves a power of a polynomial inside another power function, which makes the use of the chain rule essential. The additional constant (+7) does not affect the derivative, as constants have a derivative of zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4563d367-0a46-4d3e-a655-0e09a2a9add9%2F002e202a-db29-4ccd-9ff0-424210d575b8%2Fpx49s2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Calculating Derivative**
**Task:**
Find the derivative with respect to \( x \) of the function:
\[ f(x) = \left( (2x^3 + 4)^5 + 6 \right)^8 + 7. \]
**Solution:**
To find \( f'(x) \), apply the chain rule multiple times due to the composition of functions within the given expression.
1. Start with the outermost function and work your way inward.
2. Identify each layer of functions and differentiate accordingly, applying the chain rule.
**Comments:** The function involves a power of a polynomial inside another power function, which makes the use of the chain rule essential. The additional constant (+7) does not affect the derivative, as constants have a derivative of zero.
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