Chapter 2.1 – Limit Definition of the Derivative vs. Alternate Definition of the Derivative

The derivative is a fundamental concept in calculus, representing the instantaneous rate of change of a function or the slope of the tangent line to the function's graph at a specific point.
It extends the idea of the slope of a straight line to curves, allowing us to analyze how functions change at any given instant.

Background: For a straight line, the slope is constant and easily calculated as \frac{ ext{rise}}{ ext{run}}. For a curve, the slope changes from point to point.
Goal: To find the exact slope of a curve at a single point, which is defined as the slope of the tangent line at that point.
Secant Lines: To approach the tangent line, we consider a secant line that connects two points on the curve: (x, f(x)) and (x+h, f(x+h)). The slope of this secant line is given by: \text{m}_{sec} = \frac{f(x+h) - f(x)}{(x+h) - x} = \frac{f(x+h) - f(x)}{h} where h is the horizontal distance between the two points.
Limit Process: As the second point gets closer and closer to the first point (i.e., as h \to 0), the secant line approaches the tangent line. The slope of the tangent line is then the limit of the slopes of the secant lines.

This definition provides a formula to find the derivative of a function f(x) at any point x in its domain, resulting in a new function, f'(x), which gives the slope of the tangent line at any x.
Formula: The derivative of f(x) with respect to x is defined as:
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
Explanation of Components:
- f(x): The value of the original function at point x.
- f(x+h): The value of the original function at a point slightly to the right (or left) of x, representing a small change in x by h.
- f(x+h) - f(x): The change in the function's output (the