The derivative of a function represents the rate at which the function's value changes with respect to a variable. It provides the slope of the tangent line to the function's graph at any point. In calculus, derivatives are fundamental for understanding changes and behaviors of functions.

Basic Derivative Rules:

1. Constant Rule:
   If $f(x) = c$ (a constant), then $f'(x) = 0$.

2. Power Rule:
   If $f(x) = x^n$, then $f'(x) = n \cdot x^{n-1}$.

3. Sum Rule:
   If $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$.

4. Product Rule:
   If $f(x) = g(x) \cdot h(x)$, then $f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$.

5. Quotient Rule:
   If $f(x) = \frac{g(x)}{h(x)}$, then
   $f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{[h(x)]^2}$.

6. Chain Rule:
   If $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$.

Example:

If $f(x) = 3x^2 + 2x + 1$, applying the power rule, the derivative is: $f'(x) = 6x + 2$

Would you like further details on how to apply these rules or have any specific function you'd like to differentiate?

Expanding Questions:

1. How do you differentiate trigonometric functions?
2. Can you explain the physical significance of a derivative in real-world applications?
3. What is the second derivative and what does it represent?
4. How do you use the chain rule with composite functions?
5. Can derivatives be applied to implicit functions, and how?

Tip:

When taking the derivative of a product of two functions, don't forget to apply the product rule!