The derivative of a function measures how the function changes as its input changes. In other words, it represents the rate of change or slope of the function at any given point.

### Common Derivative Rules:

1. **Power Rule**:
   \[
   \frac{d}{dx} \left( x^n \right) = n \cdot x^{n-1}
   \]
   Example: \(\frac{d}{dx} \left( x^3 \right) = 3x^2\)

2. **Sum Rule**:
   \[
   \frac{d}{dx} \left( f(x) + g(x) \right) = f'(x) + g'(x)
   \]

3. **Product Rule**:
   \[
   \frac{d}{dx} \left( f(x) \cdot g(x) \right) = f'(x) \cdot g(x) + f(x) \cdot g'(x)
   \]

4. **Quotient Rule**:
   \[
   \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2}
   \]

5. **Chain Rule**:
   \[
   \frac{d}{dx} \left( f(g(x)) \right) = f'(g(x)) \cdot g'(x)
   \]

Would you like more details or a specific example worked out?

Here are 5 questions related to derivatives:
1. What is the derivative of \( \sin(x) \)?
2. How do you apply the chain rule for \( \cos(3x) \)?
3. What is the derivative of \( e^x \)?
4. How does the product rule apply to \( x^2 \sin(x) \)?
5. Can you explain the quotient rule with an example?

**Tip**: When differentiating, always simplify the expression before and after applying derivative rules for better clarity!

Explore the key concepts of derivatives in calculus, including the power rule, product rule, quotient rule, and chain rule. This detailed guide is suitable for high school and college-level students studying calculus.

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