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Increment Theorem and Proof


Let y = f (x). Suppose f'(x) exists at a certain point x, and Δx is infinitesimal. Then Δy is infinitesimal, and

Δy = f'(x) Δx + ε Δx for some infinitesimal ε, which depends on x and Δx.


Case 1 Δx = 0. In this case, Δy = f'(x) Δx = 0, and we put ε = 0.

Case 2 Δx ≠ 0. Then


so for some infinitesimal ε,


Multiplying both sides by Δx,

Δy = f '(x) Δx + ε Δx.

Last Update: 2010-11-25