AP Calculus AB and BC glossary

Alternate form of the derivative

The alternate form defines the derivative at a as the limit of f(x) minus f(a) over x minus a as x approaches a. It is equivalent to the h version, and it is the form to recognize when an exam gives you a limit and asks what derivative it represents.

f′(a)=lim⁡x→af(x)−f(a)x−af'(a) = \lim_{x \to a}\frac{f(x) - f(a)}{x - a}

A limit such as lim⁡x→3x5−243x−3\lim_{x \to 3}\frac{x^{5} - 243}{x - 3} is not a hard limit problem, it is f′(3)f'(3) for f(x)=x5f(x) = x^{5}. Recognizing the shape turns it into 5⋅34=4055 \cdot 3^{4} = 405 with no work.

The mistake

Reaching for L'Hopital's Rule. It works, but it is circular when the limit IS the definition of the derivative you would need in order to apply it.

Appears in: Unit 2: Defining the Derivative