Proving f(x) = x² has Derivative 2x

Real Analysis · Axiom Academy

EXAMPLE Proving f(x) = x^2 has Derivative 2x Using the limit definition of the derivative to prove that Starting from the limit definition of the derivative, prove that the function f(x) = x^2 has derivative f'(x) = 2x . Work through the algebra one step at a time. For every nonzero h , the difference quotient simplifies to the same simple expression, 2x + h . As h shrinks toward 0 , that value slides to 2x — the derivative. The difference quotient is exactly 2x + h for every . The limit is what that expression approaches as , namely 2x . The figure is a fixed snapshot — nothing here moves. Nice work — you proved straight from the limit definition that the derivative of x^2 is 2x . The moves worth remembering: The limit definition is the foundation: . Algebra clears the form: expanding, canceling the x^2 terms, and factoring out h turn the quotient into the clean expression 2x + h . The condition is what lets you cancel: we are taking the limit as h approaches 0 , never setting h = 0 , so dividing by h is legal. The same first-principles method proves the derivative of any function — and this case is exactly the power rule in action.

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