Limit Definition of the Derivative

AP Calculus · Axiom Academy

LESSON Limit Definition of the Derivative Understand the formal definition that powers all of calculus To find the slope at a point, we start by drawing a secant line—a line that passes through two points on the curve. The slope of this secant line is the familiar rise-over-run formula: We can rewrite this using function notation where is the distance between the two x-values: Now for the crucial move: what if we make h infinitely small? As h approaches 0, the secant line becomes the tangent line, and its slope becomes the derivative. This gives us the formal definition of the derivative: The derivative of f(x) at x = a is: The derivative exists if this limit exists and is finite. Sometimes it's more convenient to use a slightly different notation. Instead of using h, we can let x approach a: Both definitions are equivalent. Use whichever feels more natural for the problem you're solving. There are several ways to denote the derivative of f(x): In this course, we'll primarily use and notation. You've mastered the limit definition of the derivative! Next: Learn when derivatives exist (differentiability) and how they relate to continuity.

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