1. **State the problem:** Understand what the difference quotient means in mathematics. 2. **Definition:** The difference quotient is a formula that calculates the average rate of change of a function $f(x)$ over an interval from $x=a$ to $x=a+h$. 3. **Formula:** It is given by $$\frac{f(a+h) - f(a)}{h}$$ where $h \neq 0$. 4. **Explanation:** This formula measures how much the function's output changes when the input changes by $h$. It is the foundation for the concept of the derivative in calculus. 5. **Important note:** As $h$ approaches zero, the difference quotient approaches the instantaneous rate of change (the derivative) at $x=a$. 6. **Summary:** The difference quotient is essentially the slope of the secant line between two points on the graph of $f(x)$, representing the average rate of change over that interval.