Proving the Product Rule

Real Analysis · Axiom Academy

EXAMPLE Proving the Product Rule Using the add-and-subtract trick to rigorously prove the derivative of a product from the limit definition. Let f and g be differentiable at x . Prove the Product Rule : , starting from the limit definition of the derivative. Picture fg as the area of a rectangle with sides f and g . Nudging the inputs grows each side by and , so the area gains three pieces: , , and the tiny corner . Dividing by h and letting , the corner term shrinks faster than the rest and disappears — leaving exactly . Nice work — you built a full, rigorous proof of the Product Rule. Here is what makes it run: The add-and-subtract trick is the key insight: inserting (and subtracting it) costs nothing but creates terms you can factor. Strategic grouping reveals structure: the four terms split into two pairs, each holding a difference quotient. Continuity is essential: because f is differentiable it is continuous, so . Limit laws justify every split: the sum and product rules for limits let you break the expression apart safely. The add-and-subtract move is a workhorse of analysis — the same idea proves the quotient rule and many other limit results by manufacturing structure where none is obvious.

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