Slope is the rate at which a line rises or falls per unit of horizontal change. In AP Calculus, slope is the bridge to the derivative, since the derivative of a function at a point equals the slope of the tangent line to its graph at that point (CED 2.2.B).
Slope is the steepness of a line, measured as change in y over change in x. You've computed it since algebra as rise over run. AP Calculus takes that one idea and turns it into the entire subject of differentiation.
Here's the upgrade. The slope of a secant line through two points on a curve gives you the average rate of change of the function on that interval, which is the difference quotient (f(b) − f(a))/(b − a) from CED 2.1.A. When you let those two points slide together by taking the limit as h → 0, the secant line becomes the tangent line, and its slope is the instantaneous rate of change, also known as the derivative f'(a) (CED 2.1.B and 2.2.A). So when your teacher says "the derivative is just a slope," they mean it literally. Every derivative you compute in AP Calc is the slope of some tangent line.
Slope is the foundation of Unit 2 (Differentiation: Definition and Fundamental Properties). Learning objective 2.2.B says it outright: the derivative of a function at a point is the slope of the tangent line at that point. Topics 2.1 through 2.3 build that idea step by step, from average rates of change (secant slopes) to instantaneous rates (tangent slopes) to estimating derivatives from tables and graphs (2.3.A).
Then slope comes roaring back in Unit 7 (Differential Equations). A differential equation like dy/dx = x(y − 2)² is literally a formula that hands you the slope at every point in the plane. Verifying solutions (7.2.A), drawing slope fields, and running Euler's method (7.5.A) all depend on reading dy/dx as "the slope right here." If slope clicks, half of AP Calc clicks with it.
Keep studying AP Calculus Unit 2
Visual cheatsheet
view galleryDifference Quotient (Unit 2)
The difference quotient (f(a+h) − f(a))/h is just the slope formula in disguise, with the two points written as (a, f(a)) and (a+h, f(a+h)). Take its limit as h → 0 and you get the derivative.
Slope of the Tangent Line (Unit 2)
This is the headline fact of CED 2.2.B. Finding f'(a) and finding the tangent slope at x = a are the same task, which is why tangent line equations y − f(a) = f'(a)(x − a) show up constantly on the exam.
Solution Curve (Unit 7)
A differential equation tells you the slope dy/dx at every point, and a solution curve is any curve whose slope matches that rule everywhere. Verifying a solution (7.2.A) means plugging in and checking the slopes agree.
Initial Condition (Unit 7)
Euler's method (7.5.A) is slope put to work. You start at an initial condition, follow the slope from the differential equation for a small step, recompute the slope, and repeat to approximate the solution curve.
Multiple-choice questions ask you to compute secant slopes from a formula or table, estimate a derivative from a graph (CED 2.3.A), or pick the slope field that matches a differential equation. A classic stem looks like the practice question "find the slope of the secant line through f(x) = x³ + 2x − 4 on [−1, 2]," which is just (f(2) − f(−1))/(2 − (−1)).
On FRQs, slope shows up everywhere. The 2017 FRQ Q4 (cooling potato) and 2021 FRQ Q6 (medication) both gave differential equations where dy/dt is a slope you evaluate and interpret with units. The 2018 FRQ Q6 asked about dy/dx = x(y − 2)², where sketching the slope field and running Euler's method both require treating the equation as a slope generator. Expect to compute a slope, write a tangent line, and explain what the slope means in context, all in one problem.
A secant slope uses two points and gives the average rate of change over an interval (CED 2.1.A). A tangent slope uses one point and gives the instantaneous rate of change, which is the derivative f'(a) (CED 2.2.B). The tangent slope is the limit of secant slopes as the two points squeeze together. On the exam, the words 'average' and 'on the interval' signal secant; 'at the point' and 'instantaneous' signal tangent.
Slope is change in y per unit change in x, and in calculus it becomes the rate of change of a function.
The derivative of a function at a point equals the slope of the tangent line to the graph at that point (CED 2.2.B).
The slope of a secant line through (a, f(a)) and (b, f(b)) is the average rate of change, (f(b) − f(a))/(b − a).
Taking the limit of secant slopes as h → 0 gives the instantaneous rate of change, which is the formal definition of the derivative.
You can estimate slope (and therefore the derivative) from a table or graph by computing rise over run between nearby points (CED 2.3.A).
In Unit 7, a differential equation assigns a slope to every point, and slope fields and Euler's method are just ways of following those slopes.
Slope is the steepness of a line, computed as change in y over change in x. AP Calc extends it to curves, where the slope of the tangent line at a point is the derivative f'(a), the instantaneous rate of change.
Yes, at a point. CED 2.2.B states that the derivative of a function at a point is the slope of the line tangent to the graph at that point. The derivative function f'(x) just packages the tangent slope at every x into one formula.
A secant slope uses two points and measures the average rate of change over an interval, like the slope of 7 − 8x on [3, 10]. A tangent slope uses one point and measures the instantaneous rate of change, found by taking the limit of secant slopes as the interval shrinks to zero.
Pick the two table values closest to your point and compute (f(b) − f(a))/(b − a). That secant slope is your estimate of the derivative, which is exactly what CED 2.3.A asks you to do.
A differential equation like dy/dx = x(y − 2)² (from the 2018 FRQ) is a rule giving the slope at every point. Slope fields visualize those slopes, and Euler's method follows them step by step from an initial condition to approximate a solution curve.