Let Calculate the derivative using the product rule. f(x) = and g(x) = which means that f'(x) = and g'(x) = = And thus y = + (It doesn't matter which part of the sum you enter first or second.) Try checking your answer by multiplying y out and then using power rule. y = √√x(x² + x³)
Let Calculate the derivative using the product rule. f(x) = and g(x) = which means that f'(x) = and g'(x) = = And thus y = + (It doesn't matter which part of the sum you enter first or second.) Try checking your answer by multiplying y out and then using power rule. y = √√x(x² + x³)
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
![Let
\[ y = \sqrt[3]{x(x^2 + x^5)} \]
Calculate the derivative using the product rule.
\[ f(x) = \square \quad \text{and} \quad g(x) = \square \]
which means that
\[ f'(x) = \square \quad \text{and} \quad g'(x) = \square \]
And thus
\[ y' = \square \cdot \square + \square \cdot \square \]
(It doesn't matter which part of the sum you enter first or second.)
Try checking your answer by multiplying \( y \) out and then using the power rule.
---
Explanation:
This instructional module outlines the steps to find the derivative of a composite function using the product rule. The primary function presented is:
\[ y = \sqrt[3]{x(x^2 + x^5)} \]
The task involves identifying functions \( f(x) \) and \( g(x) \), and then computing their derivatives \( f'(x) \) and \( g'(x) \). The final derivative \( y' \) is then obtained by applying the product rule. The empty boxes are placeholders to be filled by students with the appropriate expressions and derivatives.
The product rule formula implemented here is:
\[ (f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x) \]
The instructional note advises students to simplify the function \( y \) first and then verify the derivative using the power rule as an alternative method for confirmation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F461e6b03-e26d-48fc-aa6c-204c3b542fc6%2Fa29d4c2e-3a7d-45f1-b898-1badef7809df%2Fm55fswr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let
\[ y = \sqrt[3]{x(x^2 + x^5)} \]
Calculate the derivative using the product rule.
\[ f(x) = \square \quad \text{and} \quad g(x) = \square \]
which means that
\[ f'(x) = \square \quad \text{and} \quad g'(x) = \square \]
And thus
\[ y' = \square \cdot \square + \square \cdot \square \]
(It doesn't matter which part of the sum you enter first or second.)
Try checking your answer by multiplying \( y \) out and then using the power rule.
---
Explanation:
This instructional module outlines the steps to find the derivative of a composite function using the product rule. The primary function presented is:
\[ y = \sqrt[3]{x(x^2 + x^5)} \]
The task involves identifying functions \( f(x) \) and \( g(x) \), and then computing their derivatives \( f'(x) \) and \( g'(x) \). The final derivative \( y' \) is then obtained by applying the product rule. The empty boxes are placeholders to be filled by students with the appropriate expressions and derivatives.
The product rule formula implemented here is:
\[ (f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x) \]
The instructional note advises students to simplify the function \( y \) first and then verify the derivative using the power rule as an alternative method for confirmation.
![# Calculating Derivatives Using the Product Rule
## Problem Statement
Let
\[ y = 5x^3(x^7 - 5) \]
Calculate the derivative using the product rule.
## Steps to Follow
### Step 1: Identify Functions
Let \( f(x) \) and \( g(x) \) be functions such that their product is equal to \( y \).
\[ f(x) = \hspace{48mm} g(x) = \]
### Step 2: Derivatives of the Identified Functions
Differentiate both \( f(x) \) and \( g(x) \) with respect to \( x \).
\[ f'(x) = \hspace{48mm} g'(x) = \]
### Step 3: Apply the Product Rule
Using the product rule for differentiation, which states:
\[ (f(x) \cdot g(x))' = f'(x)g(x) + f(x)g'(x) \]
We can find the derivative \( y' \):
\[ y' = \hspace{48mm} + \hspace{48mm} \]
(Note: It does not matter which part of the sum you enter first or second.)
## Verification Step
Try checking your answer by multiplying \( y \) out and then using the power rule for differentiation.
## Conclusion
By following these steps and using the product rule, you can calculate the derivative of the given function accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F461e6b03-e26d-48fc-aa6c-204c3b542fc6%2Fa29d4c2e-3a7d-45f1-b898-1badef7809df%2Fg5im8ii_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Calculating Derivatives Using the Product Rule
## Problem Statement
Let
\[ y = 5x^3(x^7 - 5) \]
Calculate the derivative using the product rule.
## Steps to Follow
### Step 1: Identify Functions
Let \( f(x) \) and \( g(x) \) be functions such that their product is equal to \( y \).
\[ f(x) = \hspace{48mm} g(x) = \]
### Step 2: Derivatives of the Identified Functions
Differentiate both \( f(x) \) and \( g(x) \) with respect to \( x \).
\[ f'(x) = \hspace{48mm} g'(x) = \]
### Step 3: Apply the Product Rule
Using the product rule for differentiation, which states:
\[ (f(x) \cdot g(x))' = f'(x)g(x) + f(x)g'(x) \]
We can find the derivative \( y' \):
\[ y' = \hspace{48mm} + \hspace{48mm} \]
(Note: It does not matter which part of the sum you enter first or second.)
## Verification Step
Try checking your answer by multiplying \( y \) out and then using the power rule for differentiation.
## Conclusion
By following these steps and using the product rule, you can calculate the derivative of the given function accurately.
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