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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Question
![The problem presented asks to find the derivative of the function \( f(x) = (5x - 1)^2 \).
\[ \text{Find } f'(x) \text{ if } f(x) = (5x - 1)^2. \]
### Solution
To find \( f'(x) \), apply the chain rule, which is expressed as:
\[ (g(x)^n)' = n \cdot g(x)^{n-1} \cdot g'(x). \]
For this function:
1. Let \( g(x) = 5x - 1 \) and \( n = 2 \).
2. Then \( g'(x) = 5 \).
Thus, applying the chain rule:
\[ f'(x) = 2 \cdot (5x - 1)^{2-1} \cdot 5. \]
Simplifying, we get:
\[ f'(x) = 2 \cdot (5x - 1) \cdot 5 = 10 \cdot (5x - 1). \]
Therefore, the derivative \( f'(x) \) is:
\[ f'(x) = 50x - 10. \]
\( f'(x) = \) \[ \boxed{50x - 10} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00318191-7541-4e8b-81d4-6b3dcc212ad8%2F31d0dfed-b45b-4bbd-8b6d-28ec5d291ca6%2F9ddkp6o_processed.png&w=3840&q=75)
Transcribed Image Text:The problem presented asks to find the derivative of the function \( f(x) = (5x - 1)^2 \).
\[ \text{Find } f'(x) \text{ if } f(x) = (5x - 1)^2. \]
### Solution
To find \( f'(x) \), apply the chain rule, which is expressed as:
\[ (g(x)^n)' = n \cdot g(x)^{n-1} \cdot g'(x). \]
For this function:
1. Let \( g(x) = 5x - 1 \) and \( n = 2 \).
2. Then \( g'(x) = 5 \).
Thus, applying the chain rule:
\[ f'(x) = 2 \cdot (5x - 1)^{2-1} \cdot 5. \]
Simplifying, we get:
\[ f'(x) = 2 \cdot (5x - 1) \cdot 5 = 10 \cdot (5x - 1). \]
Therefore, the derivative \( f'(x) \) is:
\[ f'(x) = 50x - 10. \]
\( f'(x) = \) \[ \boxed{50x - 10} \]
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