The Derivative: Formal Definition

Calculus 1 · Axiom Academy

LESSON The Derivative: Formal Definition From the slope of a secant line to the instantaneous rate of change — discovering the fundamental formula of calculus. Pick a fixed point on a curve at x , and a second point a small distance h away at x + h . The straight line joining them is the secant line , and its slope is just rise over run. The rise is the change in height, f(x + h) − f(x) , and the run is the horizontal gap, h . So the slope of the secant line is: What happens as h shrinks toward zero? The second point slides closer and closer to the first, the secant line pivots, and its slope homes in on a single value — the slope of the tangent line . The slope of the tangent line is the limit of the secant slopes as h approaches zero: Putting the secant slope and the limit together gives the definition of the derivative. Watch the secant collapse onto the tangent and its slope settle on one exact number. The derivative of a function f at a point x is This single formula encapsulates: The slope of the tangent line at any point The instantaneous rate of change of the function The velocity at any instant (for a position function) Mathematicians write the derivative several ways, each emphasizing a different aspect. Watch the tangent slide along the curve — all three notations report the same slope at every point. Key idea: the derivative f'(x) is itself a function — it returns the slope of the tangent line at every point x in the domain of f .

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