AP Calculus AB and BC glossary

Limit definition of the derivative

Also called: Definition of the derivative, Formal definition of the derivative

The limit definition of the derivative sets f prime of x equal to the limit, as h approaches zero, of the difference quotient: f of x plus h minus f of x, all over h. It turns the average rate of change over a shrinking interval into the instantaneous rate of change at a point.

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}

The difference quotient f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h} is the expression inside; on its own it is an average rate of change over an interval of width hh. Wrapping it in the limit collapses that interval to a point and returns the instantaneous rate. The alternate form f′(a)=lim⁡x→af(x)−f(a)x−af'(a) = \lim_{x \to a}\frac{f(x) - f(a)}{x - a} says the same thing at a fixed point aa.

The exam often hides a derivative inside a limit and asks you to name it. Reading lim⁡h→0sin⁡(π6+h)−sin⁡π6h\lim_{h \to 0}\frac{\sin\left(\frac{\pi}{6} + h\right) - \sin\frac{\pi}{6}}{h} as f′(π6)f'\left(\frac{\pi}{6}\right) for f(x)=sin⁡xf(x) = \sin x turns a hard limit into cos⁡π6=32\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}.

The mistake

On a recognize-the-derivative problem, pairing the wrong function or point. In lim⁡h→0ln⁡(2+h)−ln⁡2h\lim_{h \to 0}\frac{\ln(2 + h) - \ln 2}{h} the pieces force f(x)=ln⁡xf(x) = \ln x and a=2a = 2, so the value is f′(2)=12f'(2) = \frac{1}{2}, not something found by plugging h=0h = 0 into 00\frac{0}{0}.

Appears in: Unit 2: Defining the Derivative