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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![**Problem: Differentiation**
Objective: Find \(\frac{dy}{dx}\).
Given:
\[ y = \sec x - 7\sqrt{x} + 9 \]
Instructions: Calculate the derivative of the function \(y\) with respect to the variable \(x\).
---
**Explanation:**
1. Derivative of \( \sec x \):
- Using the derivative formula, \(\frac{d}{dx}[\sec x] = \sec x \tan x\).
2. Derivative of \(-7\sqrt{x}\):
- Rewrite the expression as \(-7x^{1/2}\) to make differentiation easier.
- Use the power rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\).
- Applying the power rule, \(\frac{d}{dx}[-7x^{1/2}] = -7 \cdot \frac{1}{2}x^{-1/2} = -\frac{7}{2\sqrt{x}}\).
3. Derivative of the constant \(9\):
- The derivative of any constant is \(0\).
Bringing it all together:
\[
\frac{dy}{dx} = \sec x \tan x - \frac{7}{2\sqrt{x}}
\]
Final Expression:
- The derivative of the given function is \(\frac{dy}{dx} = \sec x \tan x - \frac{7}{2\sqrt{x}}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff17eecc8-d5a6-40f8-843b-f14b601b8c9d%2F95c00f1c-91d0-43f0-84a2-a82054413805%2Fxbbqaj8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Differentiation**
Objective: Find \(\frac{dy}{dx}\).
Given:
\[ y = \sec x - 7\sqrt{x} + 9 \]
Instructions: Calculate the derivative of the function \(y\) with respect to the variable \(x\).
---
**Explanation:**
1. Derivative of \( \sec x \):
- Using the derivative formula, \(\frac{d}{dx}[\sec x] = \sec x \tan x\).
2. Derivative of \(-7\sqrt{x}\):
- Rewrite the expression as \(-7x^{1/2}\) to make differentiation easier.
- Use the power rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\).
- Applying the power rule, \(\frac{d}{dx}[-7x^{1/2}] = -7 \cdot \frac{1}{2}x^{-1/2} = -\frac{7}{2\sqrt{x}}\).
3. Derivative of the constant \(9\):
- The derivative of any constant is \(0\).
Bringing it all together:
\[
\frac{dy}{dx} = \sec x \tan x - \frac{7}{2\sqrt{x}}
\]
Final Expression:
- The derivative of the given function is \(\frac{dy}{dx} = \sec x \tan x - \frac{7}{2\sqrt{x}}\).
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