https://file.aiquickdraw.com/mathbot/file/17274314358374w6mxkk9.jpg

The problem asks to find the derivative of the function \( g(x) = \frac{x^3 - x}{x^2 + 1} \) using the quotient rule. The quotient rule is given by:

\[
\frac{d}{dx} \left( \frac{f(x)}{h(x)} \right) = \frac{f'(x)h(x) - f(x)h'(x)}{(h(x))^2}
\]

Here, the numerator is \( f(x) = x^3 - x \) and the denominator is \( h(x) = x^2 + 1 \).

### Step-by-step solution:
1. **Find \( f'(x) \)**:
   \[
   f(x) = x^3 - x
   \]
   \[
   f'(x) = 3x^2 - 1
   \]

2. **Find \( h'(x) \)**:
   \[
   h(x) = x^2 + 1
   \]
   \[
   h'(x) = 2x
   \]

3. **Apply the quotient rule**:
   \[
   g'(x) = \frac{(3x^2 - 1)(x^2 + 1) - (x^3 - x)(2x)}{(x^2 + 1)^2}
   \]

4. **Simplify the numerator**:
   First, expand \( (3x^2 - 1)(x^2 + 1) \):
   \[
   (3x^2 - 1)(x^2 + 1) = 3x^4 + 3x^2 - x^2 - 1 = 3x^4 + 2x^2 - 1
   \]

   Now, expand \( (x^3 - x)(2x) \):
   \[
   (x^3 - x)(2x) = 2x^4 - 2x^2
   \]

   Subtract the two expressions:
   \[
   3x^4 + 2x^2 - 1 - (2x^4 - 2x^2) = 3x^4 + 2x^2 - 1 - 2x^4 + 2x^2 = x^4 + 4x^2 - 1
   \]

5. **Final expression**:
   \[
   g'(x) = \frac{x^4 + 4x^2 - 1}{(x^2 + 1)^2}
   \]

Thus, the derivative of \( g(x) \) is:

\[
g'(x) = \frac{x^4 + 4x^2 - 1}{(x^2 + 1)^2}
\]

Would you like further clarification or details?

Here are 5 related questions to extend this concept:
1. How does the quotient rule differ from the product rule?
2. What happens if the function had a higher degree polynomial in both the numerator and the denominator?
3. Can you apply the chain rule to find the derivative of \( g(x) \)?
4. What would the derivative of the same function look like if calculated without the quotient rule?
5. How would you interpret the derivative \( g'(x) \) in terms of the slope of the function?

**Tip**: When simplifying the results from the quotient rule, expand terms carefully and combine like terms step by step to avoid errors.

Find the derivative of g(x) = (x^3 - x) / (x^2 + 1) using the quotient rule.

Learn how to find the derivative of the function g(x) = (x^3 - x) / (x^2 + 1) using the quotient rule. This step-by-step solution involves calculus concepts such as the quotient rule, differentiation, and simplifying expressions. Suitable for advanced high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation