AP Calculus AB and BC

Unit 2: Differentiation: Definition and Fundamental Properties

Exam weighting: AB 10-15% · BC 5-10%

Unit 2 turns Unit 1's limits into the derivative. It defines f'(x) as the limit of the difference quotient, ties it to the tangent-line slope, then gives the shortcut rules: power, sum, constant multiple, product, and quotient, plus derivatives of sin x, cos x, e^x, ln x, and the trig functions. Worth 10-15% of AB.

The big idea

The derivative is a single limit wearing three costumes: the instantaneous rate of change of ff, the slope of the line tangent to the graph, and a new function f′(x)f'(x) you can evaluate anywhere. Unit 2 defines that limit, then builds fast rules so you never have to compute it from scratch again.

Everything starts from the difference quotient. The average rate of change of ff over an interval is f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}, equivalently f(x)−f(a)x−a\frac{f(x)-f(a)}{x-a} (Topic 2.1). Let the interval shrink toward zero and, when the limit exists, you land on the instantaneous rate of change: the derivative at aa. The same limit taken at a general input defines the derivative as a function, f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} (Topic 2.2).

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

The unit moves in three stages. Topics 2.1 to 2.3 are about meaning: define the derivative, read it off graphs and tables, and estimate it numerically. Topic 2.4 draws the line between differentiability and continuity: differentiable forces continuous, but continuous does not force differentiable. Topics 2.5 to 2.10 are the toolkit: the power rule, the linearity rules, the named derivatives of sin⁡x\sin x, cos⁡x\cos x, exe^x, and ln⁡x\ln x, and finally the product, quotient, and remaining trig rules.

This is where method selection starts to earn points. Before you differentiate, name the structure of the expression: is it a sum of terms (differentiate term by term, Topic 2.6), a product (Topic 2.8), or a quotient (Topic 2.9)? Rewrite when it is cheaper: a radical becomes x1/2x^{1/2} so the power rule applies, and the Topic 2.10 functions (tan⁡x\tan x, sec⁡x\sec x, and the rest) are easiest through their known results or by rewriting via sin⁡x\sin x and cos⁡x\cos x. Picking the rule is the skill the exam rewards; running the rule is the easy part.

TopicWhen you seeRuleDerivative
2.5xrx^rPower rulerxr−1rx^{r-1}
2.6sum, difference, or c⋅fc\cdot fLinearitydifferentiate term by term
2.7sin⁡x, cos⁡x, ex, ln⁡x\sin x,\ \cos x,\ e^x,\ \ln xNamed resultscos⁡x, −sin⁡x, ex, 1x\cos x,\ -\sin x,\ e^x,\ \tfrac{1}{x}
2.8a product fgfgProduct rulef′g+fg′f'g + fg'
2.9a quotient fg\tfrac{f}{g}Quotient rulef′g−fg′g2\tfrac{f'g - fg'}{g^2}
2.10tan⁡x, cot⁡x, sec⁡x, csc⁡x\tan x,\ \cot x,\ \sec x,\ \csc xRewrite via identitiessec⁡2x, −csc⁡2x, sec⁡xtan⁡x, −csc⁡xcot⁡x\sec^2 x,\ -\csc^2 x,\ \sec x\tan x,\ -\csc x\cot x

On the AB exam Unit 2 is 10-15% of the multiple-choice section (5-10% on BC). Expect multiple-choice items that hand you a table or graph and ask for f′f' at a point (Topic 2.3), questions that test whether a function is differentiable at a corner or a vertical tangent (Topic 2.4), and steady computation with the product and quotient rules. Watch for a disguised derivative: a limit such as lim⁡x→0ex−1x\lim_{x \to 0} \frac{e^x - 1}{x} is just f′(0)f'(0) for f(x)=exf(x)=e^x (Topic 2.7), so recognizing the difference-quotient pattern turns a hard limit into a one-line answer.

Topics in this unit

Topic numbers and titles from the College Board Course and Exam Description.

  • 2.1Defining Average and Instantaneous Rates of Change at a Point
  • 2.2Defining the Derivative of a Function and Using Derivative Notation
  • 2.3Estimating Derivatives of a Function at a Point
  • 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
  • 2.5Applying the Power Rule
  • 2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple
  • 2.7Derivatives of cos x, sin x, e^x, and ln x
  • 2.8The Product Rule
  • 2.9The Quotient Rule
  • 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

How to study this unit

  • Learn both forms of the definition (Topics 2.1, 2.2) and recognize the pattern backward: a limit like $\lim_{x \to 0} \frac{e^x-1}{x}$ is really $f'(0)$ for $f(x)=e^x$ (Topic 2.7).
  • Differentiability implies continuity, but not the reverse (Topic 2.4). Memorize the two ways it fails: a corner where the left and right difference quotients disagree, like $|x|$ at $x=0$, and a vertical tangent with no slope, like $\sqrt[3]{x}$ at $x=0$. If $a$ is not in the domain of $f$, it is not in the domain of $f'$.
  • Classify structure before you compute. Ask whether the expression is a sum (differentiate term by term, Topic 2.6), a product (Topic 2.8), or a quotient (Topic 2.9), then apply the matching rule instead of reaching for the hardest one by reflex.
  • Rewrite to dodge work. For the power rule (Topic 2.5), turn radicals and reciprocals into exponents first: $\sqrt{x}=x^{1/2}$ and $\frac{1}{x^2}=x^{-2}$. For Topic 2.10, rewrite $\tan x$, $\sec x$, and the rest through $\sin x$ and $\cos x$, or use the known derivatives directly.
  • Practice reading derivatives off tables and graphs (Topic 2.3). From a table, estimate $f'(a)$ with a symmetric difference quotient $\frac{f(a+h)-f(a-h)}{2h}$; from a graph, read the slope of the tangent line. Remember the calculator can return a numerical derivative at a point.

Guides for this unit

Tools and tables for this unit