Unit 2: Differentiation: Definition and Fundamental Properties
Unit 2 turns the limit into the derivative. It covers average and instantaneous rates of change, the definition of the derivative, the notation you must use, differentiability and its connection to continuity, and the basic differentiation rules: power, constant, sum, difference, constant multiple, product, quotient, and the derivatives of the six trig functions, ex, and ln x.
How to use this guide
Read it in order the first time because the ideas build. Rates of change motivate the derivative, the derivative needs notation, notation needs the rules, and the rules need practice. Work the examples with a pencil. The practice questions at the end target the exact mistakes students make most often on this unit.
After the first read, use the trap boxes and the rules table to drill the distinctions. Finish with the practice questions, then complete the recall check on the last page out loud and note any items you cannot explain yet.
What this unit is worth. Differentiation fundamentals are about 10 to 12 percent of the AP Calculus BC exam, but the real weight is larger. Every later unit uses these rules. If the power rule or the product rule feels shaky now, the chain rule, related rates, and everything after will feel shaky too.
2.1 Defining Average and Instantaneous Rates of Change at a Point
The average rate of change of f over the interval [a, b] is [f(b) − f(a)] / (b − a). On a graph it is the slope of the secant line through the two points (a, f(a)) and (b, f(b)). For f(x) = x2 over [1, 3], the average rate is (9 − 1) / (3 − 1) = 8 / 2 = 4.
The instantaneous rate of change at a point c is what you get when the interval shrinks down to c. On a graph it is the slope of the tangent line through (c, f(c)). The tangent line touches the curve at that single point and follows its direction there. Its slope is f′(c), the derivative at c.
Trap. The secant slope and the tangent slope answer different questions. The secant gives the average over an interval. The tangent gives the rate at one point. For f(x) = x2, the average over [1, 3] is 4, but the instantaneous rate at x = 1 is 2. Do not report one when the question asks for the other.
2.2 Defining the Derivative of a Function and Using Derivative Notation
The derivative of a function at x is defined by the limit f′(x) = limh→0 [f(x + h) − f(x)] / h, when that limit exists. It is the instantaneous rate of change of f at x. You should be able to write this definition from memory. Justification questions on the exam expect it.
Derivative notation comes in several forms that all mean the same thing: f′(x), y′, dy/dx, and d/dx[f(x)]. The dy/dx form is Leibniz notation and f′(x) is Lagrange (prime) notation. Read dy/dx as "the derivative of y with respect to x." On free-response questions, write the notation every time you differentiate. Bare algebra with no notation loses credit.
Trap. The definition is a limit, not a fraction you can cancel. Writing [f(x + h) − f(x)] / h and then setting h = 0 without taking the limit is the most common way to lose the point on a definition question. The limit is doing the work.
2.3 Estimating Derivatives of a Function at a Point
The difference quotient [f(x + h) − f(x)] / h (or [f(a + h) − f(a)] / h at a point a) is the slope of the secant line. Its limit as h → 0 is the derivative. Before the limit is taken, the difference quotient gives an estimate of the derivative at the point. Smaller values of h give better estimates.
You can also estimate a derivative from a table or a graph. From a table, compute [f(b) − f(a)] / (b − a) using the two entries closest to the point. From a graph, sketch the tangent line at the point and estimate its slope with rise over run between two readable points on your sketch.
Trap. A table estimate is only as good as the spacing of the data. If the table jumps by 0.5, your estimate of f′(2) is really an average over a wide interval. Say so if the question asks about accuracy. Do not present a table estimate as exact.
2.4 Connecting Differentiability and Continuity
A function is differentiable at c when f′(c) exists, which means the limit defining the derivative exists there. Geometrically, the graph has a well-defined tangent line at that point, not a corner or a break.
Differentiability implies continuity. If f is differentiable at c, then f is continuous at c. The converse is false. A function can be continuous but not differentiable at a point. The three ways this happens are corners (like |x| at x = 0, where the left and right slopes disagree), cusps (sharp points where the slopes blow up in opposite directions), and vertical tangents (where the slope is undefined because the tangent line is vertical).
Trap. Students reverse the implication. "Continuous" does not mean "differentiable." The absolute value function is continuous everywhere and fails to be differentiable at x = 0. When a question asks whether differentiability follows from continuity, the answer is no, and |x| is the counterexample to cite.
2.5 Applying the Power Rule
The Power Rule says that if f(x) = xn, then f′(x) = n · xn−1. Bring the exponent down in front and subtract one from the exponent. For f(x) = x5, f′(x) = 5x4. For f(x) = x3, f′(x) = 3x2.
The rule works for any real exponent n, including fractions and negatives. For f(x) = √x = x1/2, f′(x) = (1/2)x−1/2 = 1 / (2√x). For f(x) = 1/x = x−1, f′(x) = −x−2 = −1/x2. Rewrite roots and fractions as powers before differentiating.
Trap. The power rule applies to x raised to a constant power. It does not apply to a constant raised to x. The derivative of 2x is not x · 2x−1. Variable bases and variable exponents are different situations with different rules.
2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
Four rules handle the basic combinations. The Constant Rule says the derivative of any fixed value is zero: d/dx[c] = 0. The Sum Rule says d/dx[f + g] = f′ + g′. The Difference Rule says d/dx[f − g] = f′ − g′. The Constant Multiple Rule says fixed values factor out: d/dx[c · f(x)] = c · f′(x).
Used together they differentiate any polynomial term by term. For f(x) = 5x3 − 2x + 7: the constant multiple rule gives d/dx[5x3] = 5 · 3x2 = 15x2, the sum and difference rules split the terms, and the constant rule kills the 7. The result is f′(x) = 15x2 − 2.
Trap. The derivative of a constant is zero, but the constant multiple rule keeps the constant. In 5x3, the 5 stays and multiplies the derivative of x3. In the lone 7, there is no x attached, so the whole term differentiates to zero. Keep track of which constants are attached to x and which stand alone.
2.7 Derivatives of cos x, sin x, ex, and ln x
Four functions have derivatives worth memorizing cold. The derivative of sin x is cos x. The derivative of cos x is −sin x. Note the minus sign. It is the single most dropped sign in the unit.
The derivative of ex is ex. The exponential function is its own derivative, which is why it appears everywhere in growth models. The derivative of ln x is 1/x, for x > 0. The natural log is only defined for positive x, so its derivative carries that domain restriction.
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| ex | ex |
| ln x | 1/x |
Trap. The minus sign on the derivative of cos x is easy to drop under time pressure, and it flips every answer that follows. When you see cos x in a product or quotient rule, write the −sin x first, then continue. Build the habit of pausing on that sign.
2.8 The Product Rule
The Product Rule says d/dx[f · g] = f′ · g + f · g′. Differentiate the first, leave the second. Then leave the first, differentiate the second. Add the two results. The order of the two terms does not matter, but both must appear.
For f(x) = x2 · sin x: f′ = 2x and g′ = cos x, so d/dx[x2 sin x] = 2x · sin x + x2 · cos x. Neither factor is a constant, so the constant multiple rule does not apply. Both factors get differentiated, each in its own term.
Trap. The derivative of a product is not the product of the derivatives. Writing (x2)′ · (sin x)′ = 2x cos x misses half the answer. If both factors contain x, you need the product rule with both terms.
2.9 The Quotient Rule
The Quotient Rule says d/dx[f / g] = (f′ · g − f · g′) / g2. Low d-high minus high d-low, over low squared. The minus sign in the numerator is the part students get wrong most often.
For h(x) = (x2 + 1) / (x − 3): the numerator differentiates to 2x and the denominator to 1, so h′(x) = [2x(x − 3) − (x2 + 1)(1)] / (x − 3)2 = (2x2 − 6x − x2 − 1) / (x − 3)2 = (x2 − 6x − 1) / (x − 3)2.
Trap. Two details decide quotient rule questions: the minus sign in the numerator and the squared denominator. Writing (f′g + fg′) / g is wrong twice over. Say the rule out loud as you write it: derivative of top times bottom, minus top times derivative of bottom, all over bottom squared.
2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
The remaining trig derivatives come from the quotient rule, but memorize the results. The derivative of tan x is sec2 x. The derivative of cot x is −csc2 x. The derivative of sec x is sec x · tan x. The derivative of csc x is −csc x · cot x.
Notice the pattern. The derivatives of tan and sec are positive. The derivatives of cot and csc carry a minus sign, the same way cos x does. The co-functions (cos, cot, csc) are the ones with the negative sign.
| Function | Derivative |
|---|---|
| tan x | sec2 x |
| cot x | −csc2 x |
| sec x | sec x · tan x |
| csc x | −csc x · cot x |
Trap. The sec and csc derivatives are the ones students mix up because each contains the other function. The derivative of sec x is sec x times tan x. The derivative of csc x is negative csc x times cot x. If you remember that the co-functions take the minus sign, you only have to memorize two of the four.
Confusions That Cost Points
| Pair | How to keep them straight |
|---|---|
| Average vs instantaneous rate | Average is the secant slope over an interval. Instantaneous is the tangent slope at a point, which is the derivative. |
| Differentiable vs continuous | Differentiable implies continuous. Continuous does not imply differentiable. Corners, cusps, and vertical tangents break differentiability only. |
| Power rule vs exponential | xn differentiates to n·xn−1. A constant to the x power is a different rule entirely. |
| Product rule vs product of derivatives | d/dx[fg] = f′g + fg′. Both terms appear. The product of the derivatives alone is always wrong. |
| Quotient rule signs | Numerator is f′g − fg′ with a minus. Denominator is g2, squared. Both details matter. |
| Derivative of cos x | It is −sin x, with the minus sign. The co-functions cos, cot, and csc all take minus signs in their derivatives. |
Practice Questions
Original questions written for this guide in the style of the AP exam. Answers and explanations are on the next page, so complete the questions before checking them.
1. If f(x) = 3x4 − 2x + 7, then f′(x) =
- 12x3 − 2
- 12x3 − 2x
- 3x3 − 2
- 12x3 − 2 + 7
2. d/dx[x2 ln x] =
- 2x ln x + x
- 2x · (1/x)
- 2x ln x
- x2 · (1/x)
3. d/dx[sin x / x] =
- (x cos x − sin x) / x2
- (x cos x + sin x) / x2
- cos x / 1
- (cos x − sin x) / x
4. The equation of the tangent line to f(x) = x2 + 1 at x = 1 is
- y = 2x
- y = 2x + 1
- y − 1 = 2(x − 1)
- y = x + 1
Answer Key
1. A. Term by term: d/dx[3x4] = 12x3 by the constant multiple and power rules, d/dx[−2x] = −2, and d/dx[7] = 0 by the constant rule. B keeps an x on the −2 term, as if −2x differentiated to −2x. C applies the power rule incorrectly to 3x4, bringing down nothing. D fails to zero out the constant 7.
2. A. Product rule with f = x2 and g = ln x: f′g + fg′ = 2x · ln x + x2 · (1/x) = 2x ln x + x. B differentiates both factors and multiplies them, which is the product-of-derivatives mistake. C drops the second term of the product rule. D drops the first term.
3. A. Quotient rule with f = sin x and g = x: (f′g − fg′) / g2 = (cos x · x − sin x · 1) / x2. B uses a plus sign in the numerator, the most common quotient rule error. C cancels as if the derivative of a quotient were the quotient of the derivatives. D forgets to square the denominator and mishandles the numerator.
4. A. f(1) = 1 + 1 = 2 and f′(x) = 2x, so f′(1) = 2. Point-slope form gives y − 2 = 2(x − 1), which simplifies to y = 2x. B adds an extra 1, confusing the function value with the intercept. C uses the point (1, 1) instead of (1, 2), misreading f(1). D guesses a line through (0, 1) with the wrong slope.
One-Page Recall Check
- State the average rate of change formula and explain what the secant line represents.
- Write the limit definition of the derivative from memory.
- List the four derivative notations and name the Leibniz and Lagrange forms.
- Explain what the difference quotient estimates and what its limit gives.
- State the differentiability-continuity implication and give a counterexample to the converse.
- Name the three ways a continuous function can fail to be differentiable at a point.
- State the power rule and apply it to x−3 and √x.
- State the constant, sum, difference, and constant multiple rules.
- Give the derivatives of sin x, cos x, ex, and ln x from memory.
- State the product rule and use it on x3 · cos x.
- State the quotient rule and explain why the minus sign and the squared denominator matter.
- Give the derivatives of tan x, cot x, sec x, and csc x, and state the sign pattern.
- Find the tangent line to f(x) = x3 at x = 2 from scratch.
Where to go next. Turn every missed item above into flashcards and drill them spaced out over several days rather than in one sitting. In Rycal, open the Differentiation deck under AP Calculus BC. The deck covers the terms in this guide, and its practice questions target the same traps named here. If you have a test date, add it in the Test Planner. You can also start your next review with a Brain Dump, then check what you missed against this guide.
Key terms for this unit
Secant line, Tangent line, Derivative of a function, Derivative notation, Difference quotient, Differentiable, Differentiability implies continuity, Power Rule, Constant Rule, Sum Rule, Difference Rule, Constant Multiple Rule, Derivative of sin x, Derivative of cos x, Derivative of ex, Derivative of ln x, Product Rule, Quotient Rule, Derivative of tan x, Derivative of cot x, Derivative of sec x, Derivative of csc x
About this guide. Written for Rycal and aligned to the College Board AP Calculus BC course framework, Unit 2. All questions and explanations are original Rycal writing. Rycal is independent and is not affiliated with or endorsed by the College Board.