To solve derivatives of fractions (also known as rational functions), you often need to apply the **quotient rule** or other differentiation techniques, depending on the structure of the fraction. Here’s a step-by-step guide:

### 1. **Quotient Rule** for Differentiation
The quotient rule is used when you have a function in the form:

\[
\frac{f(x)}{g(x)}
\]

The derivative is given by:

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

Where:
- \( f(x) \) is the numerator.
- \( g(x) \) is the denominator.
- \( f'(x) \) and \( g'(x) \) are the derivatives of \( f(x) \) and \( g(x) \), respectively.

### 2. **Example Problem**

Let’s differentiate:

\[
y = \frac{2x^3 + 5x}{x^2 - 3}
\]

**Step 1:** Identify \( f(x) \) and \( g(x) \).

- \( f(x) = 2x^3 + 5x \)
- \( g(x) = x^2 - 3 \)

**Step 2:** Find the derivatives \( f'(x) \) and \( g'(x) \).

- \( f'(x) = 6x^2 + 5 \)
- \( g'(x) = 2x \)

**Step 3:** Apply the quotient rule.

\[
y' = \frac{(6x^2 + 5)(x^2 - 3) - (2x^3 + 5x)(2x)}{(x^2 - 3)^2}
\]

**Step 4:** Expand and simplify the expression.

You’ll multiply and combine like terms to get the simplified form of the derivative.

### 3. **Other Techniques**
- If the denominator is a constant, use basic power rules.
- If the fraction can be simplified first, do that before differentiating.

Would you like to see more detailed steps or examples?

### Related Questions
1. How do you apply the product rule alongside the quotient rule?
2. What are some common mistakes to avoid when using the quotient rule?
3. How do you differentiate when the numerator or denominator is a constant?
4. Can you apply logarithmic differentiation to simplify complex fractions?
5. What’s the role of chain rule when differentiating nested fractions?

**Tip:** Simplifying a fraction before differentiating can save time and reduce the complexity of your calculations.

Learn how to solve derivatives of fractions (rational functions) using the quotient rule and other differentiation techniques. Step-by-step guide with examples included. Suitable for advanced high school and college-level students.

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