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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![**Calculate the Derivative Using the Quotient Rule**
We are given the function:
\[ y = \frac{x^6}{\sqrt{x} + 4x^7} \]
To find the derivative of this function using the quotient rule, we define:
\[ f(x) = x^6 \]
\[ g(x) = \sqrt{x} + 4x^7 \]
The derivatives are:
\[ f'(x) = 6x^5 \]
\[ g'(x) = \frac{1}{2\sqrt{x}} + 28x^6 \]
Using the quotient rule, which states:
\[ \left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \]
Substitute \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \) into the quotient rule formula:
\[ \left[ \frac{x^6}{\sqrt{x} + 4x^7} \right]' = \frac{(\sqrt{x} + 4x^7) \cdot 6x^5 - x^6 \left(\frac{1}{2\sqrt{x}} + 28x^6\right)}{(\sqrt{x} + 4x^7)^2} \]
This derivative represents the rate of change of the function \( y \) with respect to \( x \) for the given rational expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7abb6008-f966-4f35-9397-157938ce1086%2F1e728e0f-2a61-4f43-bf52-6fe5d3cc534d%2Fdidfu9l_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculate the Derivative Using the Quotient Rule**
We are given the function:
\[ y = \frac{x^6}{\sqrt{x} + 4x^7} \]
To find the derivative of this function using the quotient rule, we define:
\[ f(x) = x^6 \]
\[ g(x) = \sqrt{x} + 4x^7 \]
The derivatives are:
\[ f'(x) = 6x^5 \]
\[ g'(x) = \frac{1}{2\sqrt{x}} + 28x^6 \]
Using the quotient rule, which states:
\[ \left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \]
Substitute \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \) into the quotient rule formula:
\[ \left[ \frac{x^6}{\sqrt{x} + 4x^7} \right]' = \frac{(\sqrt{x} + 4x^7) \cdot 6x^5 - x^6 \left(\frac{1}{2\sqrt{x}} + 28x^6\right)}{(\sqrt{x} + 4x^7)^2} \]
This derivative represents the rate of change of the function \( y \) with respect to \( x \) for the given rational expression.
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