1. **Stating the problem:** We want to understand why the derivative of a function $f$ at a point $x$, denoted $f'(x)$, is defined as the limit $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 2. **Formula explanation:** This formula calculates the instantaneous rate of change of the function at $x$. It measures how $f(x)$ changes as we make a very small change $h$ in $x$. 3. **Why the difference quotient?** The expression $\frac{f(x+h) - f(x)}{h}$ is called the difference quotient. It represents the average rate of change of $f$ over the interval from $x$ to $x+h$. 4. **Taking the limit as $h \to 0$:** To find the exact rate of change at $x$, we shrink the interval $h$ to zero. The limit captures the behavior of the average rate of change as the interval becomes infinitesimally small. 5. **Intuition:** Imagine zooming in on the graph of $f$ at $x$. As $h$ gets smaller, the secant line between $(x, f(x))$ and $(x+h, f(x+h))$ approaches the tangent line at $x$. The slope of this tangent line is the derivative. 6. **Summary:** The derivative definition uses the limit of the difference quotient to precisely capture the instantaneous rate of change of $f$ at $x$, which is fundamental in calculus for understanding how functions behave locally. This is why the derivative is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$.