AP Calculus AB and BC glossary

Derivative

Also called: First derivative

The derivative of a function at a point is its instantaneous rate of change there, defined as the limit of the difference quotient as the gap shrinks to zero. Geometrically it is the slope of the tangent line at that point.

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

Three descriptions of the derivative appear on the exam and all mean the same thing: the slope of the tangent line, the instantaneous rate of change, and the limit of average rates of change. Being able to move between them is most of what free-response questions test.

The derivative is itself a function. Evaluating it at a specific input gives a number, the slope at that one point, which is why f′(x)f'(x) and f′(2)f'(2) mean different kinds of object.

Units matter

If ff measures gallons and xx measures minutes, then f′f' is in gallons per minute. Interpretation questions award points for the units and the meaning in context, not the number alone.

Economics uses this derivative constantly without always naming it. Marginal cost is defined as the cost of one more unit, but what it actually is, is the derivative of total cost with respect to quantity, which is why a marginal cost curve is the slope of the total cost curve read off point by point: marginal cost.

Appears in: Unit 2: Defining the Derivative