The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. 
Home Differentiation Differentials and Tangent Lines Increment Theorem and Proof  
Search the VIAS Library  Index  
Increment Theorem and Proof
INCREMENT THEOREM 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. PROOF 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.


Home Differentiation Differentials and Tangent Lines Increment Theorem and Proof 