AP Calculus AB Flashcards: Defining the Derivative of a Function and Using Derivative Notation
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 11 cards to help you master important concepts.
What is the limit of a difference quotient?
It is an expression in the form of lim(h→0) [f(x+h) - f(x)] / h, which is used to formally define the derivative of a function f(x).
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What is the limit of a difference quotient?
It is an expression in the form of lim(h→0) [f(x+h) - f(x)] / h, which is used to formally define the derivative of a function f(x).
What does the derivative of a function at a specific point represent geometrically?
The derivative of a function at a point is the slope of the line tangent to the graph of the function at that point.
Explain the relationship between the derivative of a function at a point and the tangent line to the function's graph at that same point.
The derivative of the function evaluated at a specific point gives the exact slope of the line that is tangent to the function's graph at that point.
If you need to find the slope of the tangent line to the graph of f(x) at x=c, what mathematical operation must you perform?
You must find the derivative of the function, f'(x), and then evaluate it at x=c to get the value f'(c), which is the slope.
Why is the derivative often described as an "instantaneous rate of change"?
It is the limit of the average rate of change as the interval shrinks to zero, representing the rate of change at a single, specific moment.
What is the first crucial value you must determine when finding the equation of a line tangent to a curve at a given point?
You must first determine the derivative of the function at the given point, as this value represents the slope of the tangent line.
For a function defined as y = f(x), list three common notations for its derivative.
Three common notations for the derivative are dy/dx (Leibniz notation), f'(x) (prime notation), and y' (prime notation).
How is the derivative of a function represented as the limit of a difference quotient?
The derivative is represented as the limit of the difference quotient, lim(h→0) [f(x+h) - f(x)] / h, which calculates the instantaneous rate of change.
What are the four ways in which a derivative can be represented?
The derivative can be represented graphically (as the slope of a tangent line), numerically (through a table of values), analytically (with a formula), and verbally (as an instantaneous rate of change).
What does the notation dy/dx represent?
The notation dy/dx, also known as Leibniz notation, represents the derivative of the function y with respect to the variable x.
What is the formal definition of the derivative of a function f, denoted as f'(x)?
The derivative of f is the function whose value at x is defined by the limit f'(x) = lim(h→0) [f(x+h) - f(x)] / h, provided this limit exists.