Differentiation: Definition & Fundamental Properties

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 2. This unit is worth 5–10% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• The Derivative & Basic Rules14 min · 3 objectivesDefine the derivative as a limit of a difference quotient · Apply the power, constant-multiple, and sum rules · Differentiate products and quotients with the product and quotient rules
• Differentiability and Where Derivatives Fail14 min · 3 objectivesState the four ways a function can fail to be differentiable at a point · Explain why differentiability implies continuity and why the converse is false · Use one-sided derivatives to decide whether a piecewise function is differentiable at its seam
• Derivatives from Graphs and Tables14 min · 3 objectivesEstimate a derivative from tabulated values using a difference quotient · Read the sign and relative size of f′ off the graph of f, and reconstruct f′ qualitatively · Apply the product, quotient and chain rules to functions given only by a table of values

Formulas in Unit 2

Every term in Unit 2

All 22 terms we publish for Differentiation: Definition & Fundamental Properties, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Derivatives of inverse trig functions

arcsin′ = 1/√(1 − x²), arctan′ = 1/(1 + x²), arcsec′ = 1/(|x|√(x² − 1)).

Logarithmic differentiation

Take ln of both sides before differentiating. Required for a variable base raised to a variable exponent.

Derivative of an inverse function

(f⁻¹)′(b) = 1/f′(a) where f(a) = b. Locate the input a first; the formula uses f′ there, not at b.

Higher-order derivatives in motion

Position, velocity, acceleration and jerk are successive derivatives. Speed is the magnitude of velocity, not its derivative.

Difference-quotient definition

f′(x) = lim(h→0) [f(x + h) − f(x)]/h. Produces the derivative function; the form used to derive a general rule.

Alternate-form definition

f′(a) = lim(x→a) [f(x) − f(a)]/(x − a). Produces a number at one point; the form to recognize when a limit is centered at a specific value.

Recognizing a derivative in disguise

A limit of a difference of function values over the matching input difference IS a derivative. lim(h→0) [(2+h)⁵ − 32]/h is f′(2) for f = x⁵, so 80.

The constant subtracted is f(a)

The fastest way to identify a disguised derivative: check whether the constant being subtracted equals f(a) for some familiar f. If so, the problem is already solved.

When only the definition works

For a piecewise function at its seam, or a function like x²sin(1/x) at 0, the rules give nothing and the definition gives the answer.

Differentiability implies continuity

One direction only. The contrapositive is the useful tool: not continuous at a implies not differentiable at a.

Corner

One-sided derivatives finite but unequal, as for |x| at 0. Continuous, not differentiable.

Cusp

One-sided slopes running to +∞ and −∞, as for x^(2/3) at 0. Continuous, not differentiable.

Vertical tangent as a failure

Slope unbounded with the same sign from both sides, as for x^(1/3) at 0. A tangent line exists but is not a number.

Matching a piecewise function differentiably

Two conditions: values agree at the seam, and branch derivatives agree there. Two conditions is why such problems supply two unknown constants.

The smooth join is the tangent line

Joining a curve to a line differentiably makes that line the tangent at the seam. A different line means an arithmetic error.

Product rule

(fg)′ = f′g + fg′. Write the rule before substituting table values — students who substitute first drop a term.

Quotient rule order

(f/g)′ = (f′g − fg′)/g². The subtraction is not symmetric; reversing it negates the whole answer.

Which rule governs

Identify the OUTERMOST structure first. x²·cos(5x) is a product containing a composition; cos(x²eˣ) is a composition containing a product.

Derivative of a sum

Differentiation is linear, so sums split and constants factor out. There is no corresponding rule for products or quotients.

Tangent versus normal line

Tangent slope f′(a); normal slope −1/f′(a). The normal is vertical where f′(a) = 0.

Second derivative units

Units of y per unit of x squared — feet per second per second for a position function of time.

Symmetric difference quotient

f′(a) ≈ [f(a + h) − f(a − h)]/(2h). The divisor is 2h; using h doubles the estimate.

What examiners penalize here

• When a limit is written as lim(h→0) [f(a+h) − f(a)]/h, recognize it as the *definition of a derivative at a point*, f′(a). AP loves disguising a derivative as a raw limit — identifying it lets you skip the algebra and just differentiate.
• When a free-response question hands you a piecewise function with unknown constants and says "differentiable," it is asking for two equations, not one. Write the continuity equation and the slope equation, then solve the system.
• Table questions almost always ask you to justify the estimate. "f′(5) ≈ [f(7) − f(3)]/(7 − 3) = 3" earns the point; the bare number 3 often does not. Show the quotient you used.

Practice Calculus BC

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 10 units of AP Calculus BC

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Fundamental Properties
3. Unit 3 · Differentiation: Composite, Implicit & Inverse Functions
4. Unit 4 · Contextual Applications of Differentiation
5. Unit 5 · Analytical Applications of Differentiation
6. Unit 6 · Integration & Accumulation of Change
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration
9. Unit 9 · Parametric, Polar & Vector-Valued Functions
10. Unit 10 · Infinite Sequences & Series