Difficulty: Foundational | Prerequisites: Limits (Unit 1), function notation, slope of a line.
Unit 2 is where calculus properly begins. Everything in Unit 1 on limits was groundwork for this: the derivative is defined as a limit, and it gives you the instantaneous rate of change of a function at a point. If you can differentiate confidently and write tangent-line equations without hesitating, the rest of AP Calculus (optimisation, related rates, curve sketching) becomes far more manageable. You should already be comfortable evaluating limits and working with function notation before starting here.
The derivative measures how fast a function's output changes relative to its input, at a single point. You can find it using the limit definition or shortcut rules (power, quotient). Once you have the derivative at a point, you can write the equation of the tangent line using point-slope form.
Derivative
The instantaneous rate of change of a function at a point. Formally, f'(x) = lim(Δx→0) [f(x + Δx) − f(x)] / Δx. In simple terms, it tells you the slope of the curve at exactly one point, not over an interval.
Tangent line
A line that touches a curve at exactly one local point and has the same slope as the curve there. Think of it as the best straight-line approximation of the curve at that point.
Average rate of change (AROC)
The slope of the secant line between two points on a function: [f(b) − f(a)] / (b − a). In simple terms, this is the "overall" rate of change across an interval, like average speed over a journey.
Instantaneous rate of change (IROC)
The derivative evaluated at a single point. This is the rate of change at one precise moment, not spread over an interval. Think of it as your speedometer reading at one instant, rather than your trip's average speed.
Quotient rule
A rule for differentiating a fraction where both numerator and denominator are functions of x. If y = f(x)/g(x), then y' = [f'(x)·g(x) − f(x)·g'(x)] / [g(x)]². In simple terms, it is "low d-high minus high d-low, over low squared."
Power rule
If f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹. This is the most-used shortcut in differentiation.
Point-slope form
The equation y − y₁ = m(x − x₁), used to write the equation of a line when you know one point and the slope. This is the standard method for writing tangent-line equations on the AP exam.
You can determine the sign of a derivative at a point just by looking at the curve:
If the curve is rising (going uphill left to right) at the point, the slope of the tangent line is positive.
If the curve is falling (going downhill left to right), the slope is negative.
If the curve has a local maximum or minimum (a peak or valley) at the point, the tangent line is horizontal and the slope is zero.
This is a common AP multiple-choice setup: you are given a graph and asked to classify the tangent slope at a marked point.
To write the equation of the tangent line to f(x) at x = a:
Find the slope: compute f'(a).
Find the point: compute f(a) to get the ordered pair (a, f(a)).
Plug into point-slope form: y − f(a) = f'(a)(x − a).
Example (from the review): For f(x) = x² − 2x − 3 at x = 2:
f'(x) = 2x − 2, so f'(2) = 2. That is the slope.
f(2) = 4 − 4 − 3 = −3. The point is (2, −3).
Tangent line: y − (−3) = 2(x − 2), which simplifies to y = 2x − 7.
If you are given f(a) and f'(a) directly (without a formula), you skip straight to point-slope. For instance, if f(2) = 3 and f'(2) = −1, the tangent line at x = 2 is y − 3 = −1(x − 2), or y = −x + 5.
The formal definition is:
f'(x) = lim(Δx→0) [f(x + Δx) − f(x)] / Δx
On the AP exam, you may be asked to identify which expression correctly represents the derivative. The two key features to check:
The numerator must substitute (x + Δx) into the function properly (not just add Δx to the output).
The entire expression must include the limit as Δx → 0. Without the limit, it is just a difference quotient, not a derivative.
Example: For f(x) = 2x² + 4, the correct derivative expression is lim(Δx→0) [2(x + Δx)² + 4 − (2x² + 4)] / Δx. Note how (x + Δx) replaces x inside the function, and the limit is present.
Sometimes the AP exam gives you a limit and asks you to identify which function and which point it represents. The structure to match is:
f'(c) = lim(Δx→0) [f(c + Δx) − f(c)] / Δx
Look at what sits inside the brackets to identify f(x), and look at the base value being shifted by Δx to identify c.
Example: The expression lim(Δx→0) [2(−2 + Δx)³ − 2(−2 + Δx) − 2 − (−14)] / Δx represents f'(−2) where f(x) = 2x³ − 2x − 2. You can verify by checking that f(−2) = 2(−8) − 2(−2) − 2 = −16 + 4 − 2 = −14, which matches the constant being subtracted.
Average rate of change over [a, b] is [f(b) − f(a)] / (b − a). This is a slope between two points.
Instantaneous rate of change at t = a is f'(a). This requires differentiating first, then substituting.
Example: For f(t) = (t³ + 2)/t, the average rate of change on [1, 4] is [f(4) − f(1)] / (4 − 1) = [(66/4) − (3/1)] / 3 = [16.5 − 3] / 3 = 9/2.
For the instantaneous rate at t = 2, rewrite f(t) = t² + 2t⁻¹, then f'(t) = 2t − 2t⁻², so f'(2) = 4 − 1/2 = 7/2.
For y = 3x / (x² + 1):
Numerator function: 3x (derivative: 3)
Denominator function: x² + 1 (derivative: 2x)
Apply the rule: y' = [3(x² + 1) − 3x(2x)] / (x² + 1)²
Simplify: y' = (3x² + 3 − 6x²) / (x² + 1)² = (−3x² + 3) / (x² + 1)²
The simplified factored form, −3(x − 1)(x + 1) / (x² + 1)², may appear in multiple-choice answers.
Rule | Formula |
|---|---|
Power rule | d/dx [xⁿ] = n·xⁿ⁻¹ |
Constant multiple | d/dx [c·f(x)] = c·f'(x) |
Sum/Difference | d/dx [f(x) ± g(x)] = f'(x) ± g'(x) |
Quotient rule | d/dx [f/g] = [f'g − fg'] / g² |
Limit definition | f'(x) = lim(Δx→0) [f(x+Δx) − f(x)] / Δx |
AROC | [f(b) − f(a)] / (b − a) |
Tangent line | y − f(a) = f'(a)(x − a) |
Derivatives show up whenever you need a rate at an exact instant. A speedometer reading is the derivative of your position function. In economics, marginal cost is the derivative of the total cost function, telling a business how much one more unit will cost to produce. Any time someone says "the rate at which something is changing right now," they are talking about a derivative.
Students frequently forget the limit in the definition of the derivative and write just the difference quotient. Without lim(Δx→0), you have an average rate of change, not a derivative.
When using the quotient rule, a common error is writing the numerator as fg' − f'g instead of f'g − fg'. The order matters. A useful mnemonic: "low d-high minus high d-low."
Students often substitute Δx into the wrong place in the limit definition, adding it to the output rather than replacing x with (x + Δx) inside the function.
Confusing "the slope at a point" with "the value of the function at a point." f'(a) gives the slope of the tangent; f(a) gives the y-coordinate of the point on the curve.
⚠️ Tangent-line problems appear on nearly every AP exam, both multiple choice and free response. Being able to write one quickly and correctly is non-negotiable.
⚠️ The limit definition of the derivative is tested conceptually. You are often asked to identify the correct limit expression rather than evaluate it by hand.
⚠️ Average rate of change vs. instantaneous rate of change is a classic comparison question on the AP exam. Know when to use the secant slope formula and when to differentiate.
⚠️ Always simplify quotient-rule answers when answering multiple choice. The AP often presents equivalent but differently factored forms.
True or false: The derivative of a function at a point gives you the y-value of the function at that point.
Fill in the blank: The tangent line to f(x) at x = a has slope _______.
True or false: The expression [f(x + Δx) − f(x)] / Δx, without a limit, is a derivative.
Fill in the blank: The average rate of change of f on [2, 5] is _______.
True or false: In the quotient rule, the numerator is f'g + fg'.
Answers: 1. False (it gives the slope). 2. f'(a). 3. False (it is a difference quotient; you need lim Δx→0). 4. [f(5) − f(2)] / 3. 5. False (it is f'g − fg'; the plus sign belongs to the product rule).
Q: If f(2) = 3 and f'(2) = −1, what is the equation of the tangent line at x = 2?
A: y − 3 = −1(x − 2), which simplifies to y = −x + 5.
Q: For f(x) = x² − 2x − 3, find the equation of the tangent line at x = 2.
A: f'(x) = 2x − 2, so f'(2) = 2. f(2) = −3. Tangent line: y + 3 = 2(x − 2), or y = 2x − 7.
Q: For f(t) = (t³ + 2)/t, what is the average rate of change on [1, 4]?
A: f(4) = 66/4 = 33/2, f(1) = 3. AROC = (33/2 − 3) / 3 = (27/2) / 3 = 9/2.
Q: For f(t) = (t³ + 2)/t, what is the instantaneous rate of change at t = 2?
A: Rewrite as f(t) = t² + 2t⁻¹. Then f'(t) = 2t − 2t⁻². f'(2) = 4 − 1/2 = 7/2.
Q: Which expression represents the derivative of f(x) = 2x² + 4: (A) the difference quotient without a limit, or (C) the limit of [2(x+Δx)² + 4 − (2x² + 4)] / Δx as Δx→0?
A: (C). The derivative requires both the correct substitution of (x + Δx) into the function and the limit as Δx approaches 0.
Q: A limit expression equals lim(Δx→0) [2(−2+Δx)³ − 2(−2+Δx) − 2 − (−14)] / Δx. What are f(x) and c?
A: f(x) = 2x³ − 2x − 2 and c = −2. Verify: f(−2) = −16 + 4 − 2 = −14.
This material feeds directly into Unit 3 (chain rule, implicit differentiation) and Unit 5 (curve sketching using derivatives). The tangent-line concept returns in Unit 4 for linear approximation, where you use the tangent line to estimate function values near a known point. The average-vs.-instantaneous distinction also appears in the integral unit, where the Mean Value Theorem connects the two formally.
derivative, differentiation, tangent line, slope of tangent, point-slope form, limit definition of derivative, difference quotient, average rate of change, AROC, instantaneous rate of change, IROC, power rule, quotient rule, secant line, AP Calculus Unit 2, sections 2.1 2.2 2.3 2.4 2.5 2.6, rate of change, marginal, d/dx