Geometric Meaning of the Derivative
Big Idea: The derivative of a function at a point gives us the slope of the tangent line to the curve at that point. This is the fundamental geometric interpretation that connects algebra and geometry in calculus.
1. From Secant Lines to Tangent Lines
Consider a function $f(x)$ and two points on its graph: $A(a, f(a))$ and $B(a+h, f(a+h))$.
The secant line through $A$ and $B$ has slope:
$$m_{\text{sec}} = \frac{f(a+h) - f(a)}{h}$$
This expression is called the difference quotient. It measures the average rate of change of $f$ over the interval $[a, a+h]$.