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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![**Objective: Finding the Derivative of a Function**
**Function Given:**
\[ y = 10x^5 - \frac{2}{x^3} + 4x - 9\sqrt{x} - 10 \]
### Explanation:
The objective is to find the derivative of the function provided. The function is a combination of polynomial, rational, and radical components:
1. **Polynomial Term**: \(10x^5\)
2. **Rational Term**: \(-\frac{2}{x^3}\) (can be expressed as \(-2x^{-3}\))
3. **Linear Term**: \(4x\)
4. **Radical Term**: \(-9\sqrt{x}\) (can be expressed as \(-9x^{1/2}\))
5. **Constant Term**: \(-10\)
To find the derivative, apply the following rules:
- **Power Rule**: If \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).
- **Constant Rule**: The derivative of a constant is 0.
- **Sum/Difference Rule**: Differentiate each term separately.
### Steps:
1. Differentiate each term using the power rule.
2. Sum the derivatives of each term to get the derivative of the entire function.
### Note:
It will be helpful to rewrite the rational and radical terms using exponents before taking derivatives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a799525-bc02-4aa7-a724-19d6c1834605%2Fc4dbe439-b91e-4d16-882e-2ebe9623c94e%2Fzng660m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Objective: Finding the Derivative of a Function**
**Function Given:**
\[ y = 10x^5 - \frac{2}{x^3} + 4x - 9\sqrt{x} - 10 \]
### Explanation:
The objective is to find the derivative of the function provided. The function is a combination of polynomial, rational, and radical components:
1. **Polynomial Term**: \(10x^5\)
2. **Rational Term**: \(-\frac{2}{x^3}\) (can be expressed as \(-2x^{-3}\))
3. **Linear Term**: \(4x\)
4. **Radical Term**: \(-9\sqrt{x}\) (can be expressed as \(-9x^{1/2}\))
5. **Constant Term**: \(-10\)
To find the derivative, apply the following rules:
- **Power Rule**: If \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).
- **Constant Rule**: The derivative of a constant is 0.
- **Sum/Difference Rule**: Differentiate each term separately.
### Steps:
1. Differentiate each term using the power rule.
2. Sum the derivatives of each term to get the derivative of the entire function.
### Note:
It will be helpful to rewrite the rational and radical terms using exponents before taking derivatives.
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