dy For the following composite function, find an inner function u = g(x) and an outer function y=f(u) such that y = f(g(x)). Then calculate dx y = (4x+5)⁹ --- Select the correct choice below and fill in the answer box to complete your choice. dy d d OA. = (4x + 5) = dx du dx dy d OB. = (4x + 5) = dx du dy d d C. dx du dy d dx du O . (u). dx (4u+5). d (u®)• & (4x)= ). dx (x) =
dy For the following composite function, find an inner function u = g(x) and an outer function y=f(u) such that y = f(g(x)). Then calculate dx y = (4x+5)⁹ --- Select the correct choice below and fill in the answer box to complete your choice. dy d d OA. = (4x + 5) = dx du dx dy d OB. = (4x + 5) = dx du dy d d C. dx du dy d dx du O . (u). dx (4u+5). d (u®)• & (4x)= ). dx (x) =
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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![**Educational Website Content: Chain Rule Example**
---
### Composite Functions and the Chain Rule
For the given composite function, identify the inner function \( u = g(x) \) and the outer function \( y = f(u) \) such that \( y = f(g(x)) \). Then calculate \(\dfrac{dy}{dx}\).
**Function to Differentiate:**
\[ y = (4x + 5)^9 \]
---
**Step-by-Step Solution:**
1. **Identify Inner and Outer Functions:**
- Inner Function \( u \):
\[ u = 4x + 5 \]
- Outer Function \( y \):
\[ y = u^9 \]
2. **Apply the Chain Rule:**
The Chain Rule states:
\[ \dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx} \]
3. **Calculate the Derivatives:**
- Derivative of the outer function \( u^9 \) with respect to \( u \):
\[ \dfrac{dy}{du} = \dfrac{d}{du}(u^9) = 9u^8 \]
- Derivative of the inner function \( u = 4x + 5 \) with respect to \( x \):
\[ \dfrac{du}{dx} = \dfrac{d}{dx}(4x + 5) = 4 \]
4. **Combine Using Chain Rule:**
\[ \dfrac{dy}{dx} = (9u^8) \cdot 4 \]
\[ \dfrac{dy}{dx} = 36u^8 \]
5. **Substitute Back \( u = 4x + 5 \):**
\[ \dfrac{dy}{dx} = 36(4x + 5)^8 \]
---
**Multiple Choice Question:**
Select the correct choice below and fill in the answer box to complete your choice.
- A.
\[ \dfrac{dy}{dx} = \dfrac{d}{du}\left( u^9 \right) \cdot \dfrac{d}{dx}\left( 4x + 5 \right) = \]
- B.
\[ \d](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc944e8dc-d275-4b70-8def-6573a43e5eec%2F8b911766-92a4-4fed-b0e8-a7971f015bcc%2Ft9dpx5h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content: Chain Rule Example**
---
### Composite Functions and the Chain Rule
For the given composite function, identify the inner function \( u = g(x) \) and the outer function \( y = f(u) \) such that \( y = f(g(x)) \). Then calculate \(\dfrac{dy}{dx}\).
**Function to Differentiate:**
\[ y = (4x + 5)^9 \]
---
**Step-by-Step Solution:**
1. **Identify Inner and Outer Functions:**
- Inner Function \( u \):
\[ u = 4x + 5 \]
- Outer Function \( y \):
\[ y = u^9 \]
2. **Apply the Chain Rule:**
The Chain Rule states:
\[ \dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx} \]
3. **Calculate the Derivatives:**
- Derivative of the outer function \( u^9 \) with respect to \( u \):
\[ \dfrac{dy}{du} = \dfrac{d}{du}(u^9) = 9u^8 \]
- Derivative of the inner function \( u = 4x + 5 \) with respect to \( x \):
\[ \dfrac{du}{dx} = \dfrac{d}{dx}(4x + 5) = 4 \]
4. **Combine Using Chain Rule:**
\[ \dfrac{dy}{dx} = (9u^8) \cdot 4 \]
\[ \dfrac{dy}{dx} = 36u^8 \]
5. **Substitute Back \( u = 4x + 5 \):**
\[ \dfrac{dy}{dx} = 36(4x + 5)^8 \]
---
**Multiple Choice Question:**
Select the correct choice below and fill in the answer box to complete your choice.
- A.
\[ \dfrac{dy}{dx} = \dfrac{d}{du}\left( u^9 \right) \cdot \dfrac{d}{dx}\left( 4x + 5 \right) = \]
- B.
\[ \d
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