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...
Related questions
Question
![**Problem Statement**
Find \(\frac{dy}{dx}\) for the function:
\[ y = 9x^2 \sin x + 18x \cos x - 18 \sin x \]
---
**Solution**
To find \(\frac{dy}{dx}\), we need to differentiate each term in the function \( y \).
1. Differentiate \( y = 9x^2 \sin x \):
Apply the product rule: If \( u = 9x^2 \) and \( v = \sin x \), then
\[
\frac{d}{dx}[uv] = u'v + uv'
\]
- \( u' = \frac{d}{dx}(9x^2) = 18x \)
- \( v' = \frac{d}{dx}(\sin x) = \cos x \)
So,
\[
\frac{d}{dx}(9x^2 \sin x) = 18x \sin x + 9x^2 \cos x
\]
2. Differentiate \( y = 18x \cos x \):
Again, apply the product rule: If \( u = 18x \) and \( v = \cos x \), then
\[
\frac{d}{dx}[uv] = u'v + uv'
\]
- \( u' = \frac{d}{dx}(18x) = 18 \)
- \( v' = \frac{d}{dx}(\cos x) = -\sin x \)
So,
\[
\frac{d}{dx}(18x \cos x) = 18 \cos x - 18x \sin x
\]
3. Differentiate \( y = -18 \sin x \):
- The derivative of \(-18 \sin x\) is \(-18 \cos x\).
Combine these results:
\[
\frac{dy}{dx} = (18x \sin x + 9x^2 \cos x) + (18 \cos x - 18x \sin x) - 18 \cos x
\]
Simplify the expression:
\[
\frac{dy}{dx} = 9x^2 \cos x
\]
Therefore, the derivative of the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F256dbc8b-2bc9-463c-8f55-d3702041f8b6%2Ffbf76b91-20cb-46b2-b8f1-c25346e61cce%2Fnxbr7eo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find \(\frac{dy}{dx}\) for the function:
\[ y = 9x^2 \sin x + 18x \cos x - 18 \sin x \]
---
**Solution**
To find \(\frac{dy}{dx}\), we need to differentiate each term in the function \( y \).
1. Differentiate \( y = 9x^2 \sin x \):
Apply the product rule: If \( u = 9x^2 \) and \( v = \sin x \), then
\[
\frac{d}{dx}[uv] = u'v + uv'
\]
- \( u' = \frac{d}{dx}(9x^2) = 18x \)
- \( v' = \frac{d}{dx}(\sin x) = \cos x \)
So,
\[
\frac{d}{dx}(9x^2 \sin x) = 18x \sin x + 9x^2 \cos x
\]
2. Differentiate \( y = 18x \cos x \):
Again, apply the product rule: If \( u = 18x \) and \( v = \cos x \), then
\[
\frac{d}{dx}[uv] = u'v + uv'
\]
- \( u' = \frac{d}{dx}(18x) = 18 \)
- \( v' = \frac{d}{dx}(\cos x) = -\sin x \)
So,
\[
\frac{d}{dx}(18x \cos x) = 18 \cos x - 18x \sin x
\]
3. Differentiate \( y = -18 \sin x \):
- The derivative of \(-18 \sin x\) is \(-18 \cos x\).
Combine these results:
\[
\frac{dy}{dx} = (18x \sin x + 9x^2 \cos x) + (18 \cos x - 18x \sin x) - 18 \cos x
\]
Simplify the expression:
\[
\frac{dy}{dx} = 9x^2 \cos x
\]
Therefore, the derivative of the
Expert Solution
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