Finding f'(x) Using the Limit Definition

Calculus 1 · Axiom Academy

EXAMPLE Finding f'(x) Using the Limit Definition Differentiate from first principles — build the difference quotient, simplify, and take the limit. Find the derivative of f(x) = x^2 directly from the limit definition Nice work — you differentiated f(x) = x^2 straight from the definition and landed on f'(x) = 2x . The recipe: form , simplify until the h in the denominator cancels, then let . Why we can cancel: the h that blocked us from substituting h=0 factors out of the numerator first, so . The result: f'(x) = 2x , which agrees with the Power Rule . Every "shortcut" rule of differentiation is just this limit, worked once in general — the definition is where they all come from.

This is the written version of the interactive lesson above. See the full Calculus 1 course.