Formal Definition of Derivative

Real Analysis · Axiom Academy

LESSON The Formal Definition of the Derivative The precise limit behind the slope of a tangent line — and exactly when it exists. Let f be defined on an open interval containing a . The derivative of f at a , written f'(a) , is the limit of the slope of the secant line through and as the second point slides in toward the first. slope of the secant line — the difference quotient its limit — the slope of the tangent line The expression inside the limit is the difference quotient . Geometrically it is the slope of the secant line: a rise of f(a+h) - f(a) over a run of h . The horizontal step h — how far a+h sits from a . It shrinks toward 0 . The vertical change f(a+h) - f(a) — how much the output moves over that step. Rise over run is the average rate of change across the interval . As the secant pivots into the tangent, and the average rate becomes instantaneous . For the derivative to exist, the difference quotient must approach the same value whether from the right ( h > 0 ) or the left ( h < 0 ). When it does, that common value is f'(a) — and we can compute it directly. Worked example: f(x) = x^2 at a = 2 Form the difference quotient and simplify before letting : The cancellation is the whole trick: the troublesome h in the denominator divides out, leaving 4 + h , which has an obvious limit. The animation plots that real, computed quotient 4 + h against h . Watch it close in on the dashed line at f'(2) = 4 from both sides. 4. Differentiable Implies Continuous

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