Composite & Implicit Differentiation

Name: AP Calculus AB Unit 3: Composite & Implicit Differentiation
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9–13% of the exam4 lessons · 55 min8 terms

What this unit covers

The topics below follow the published Calculus AB course framework for Unit 3. This unit is worth 9–13% of the exam, so budget your time against that rather than against how long the unit takes to teach.

ImplicitInverse functionsHigher-orderRelated contexts

Lessons in this unit

Implicit Differentiation14 min · 3 objectivesDifferentiate both sides of an equation with respect to x, treating y as a function of x · Solve the resulting equation algebraically for dy/dx · Find the slope of a tangent line to an implicitly defined curve
Derivatives of Inverse Functions15 min · 3 objectivesUse the inverse-function relationship to compute a derivative at a point · State and apply the derivatives of the inverse trigonometric functions · Relate the slope of a function and its inverse as reciprocals at matching points
Higher-Order Derivatives12 min · 3 objectivesCompute second and higher derivatives by differentiating repeatedly · Interpret the second derivative as the rate of change of the first derivative · Connect position, velocity, and acceleration through successive derivatives
Exponential, Logarithmic & Combined Derivatives14 min · 3 objectivesDifferentiate exponential and logarithmic functions, including base-a forms · Apply the chain rule to exponential and logarithmic composites · Combine multiple differentiation rules in a single problem

Formulas in Unit 3

Chain rule on y-terms

d/dx[yⁿ] = n·yⁿ⁻¹ · (dy/dx)

Treat y as a function of x. Any term containing y produces a dy/dx factor; terms in x alone do not.

Derivative of an inverse function

If g = f⁻¹, then g′(a) = 1 / f′(g(a))

The inverse’s slope at a is the reciprocal of f’s slope at the matching point b = g(a) where f(b) = a.

Key inverse trig derivatives

d/dx[arcsin x] = 1/√(1 − x²) · d/dx[arctan x] = 1/(1 + x²)

And d/dx[arccos x] = −1/√(1 − x²). Chain rule applies for arcsin(u): u′/√(1 − u²).

Second derivative and motion

v(t) = s′(t) · a(t) = s″(t) = v′(t)

Velocity is the first derivative of position; acceleration is the second derivative of position (the derivative of velocity).

Exponential and log derivatives

d/dx[eˣ] = eˣ · d/dx[ln x] = 1/x · d/dx[aˣ] = aˣ·ln a

With the chain rule: d/dx[e^u] = e^u·u′ and d/dx[ln u] = u′/u.

Every term in Unit 3

All 8 terms we publish for Composite & Implicit Differentiation, 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.

Chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x). Differentiate the outside leaving the inside alone, then multiply by the derivative of the inside.
Implicit differentiation: Differentiate both sides with respect to x, applying the chain rule to every y term so each contributes dy/dx, then solve for dy/dx.
Recognizing a composite: Anything raised to a power, or a trig, exponential or log function of something other than plain x, needs the chain rule.
Vertical and horizontal tangents implicitly: Horizontal where the numerator of dy/dx is zero, vertical where the denominator is zero — provided the point is on the curve.
Derivative of an inverse function: (f⁻¹)′(b) = 1/f′(a) where f(a) = b. Find the input a first; the formula uses the original function's derivative there.
Derivatives of inverse trig functions: arcsin′ = 1/√(1 − x²), arccos′ = −1/√(1 − x²), arctan′ = 1/(1 + x²).
Logarithmic differentiation: Take ln of both sides before differentiating. Handles variable bases with variable exponents and long products.
Second derivative implicitly: Differentiate dy/dx again with respect to x, then substitute the expression for dy/dx wherever it appears.

What examiners penalize here

To find a tangent slope on an implicit curve you need both coordinates, because dy/dx typically depends on x *and* y. Differentiate, solve for dy/dx, then substitute the full point — not just the x-value.
The formula g′(a) = 1/f′(g(a)) shows up on the AP exam most often in table form: you are handed values of f and f′ and asked for the inverse’s derivative. Identify the matching point from the table, then take the reciprocal.
On motion problems, read the question’s verb carefully: "how fast" usually means velocity (first derivative), while "how is the velocity changing" means acceleration (second derivative). Matching the derivative order to the phrasing is worth easy points.
When several rules stack, name the outermost operation before computing anything. "This is a product, whose second factor is a composite" gives you a plan and keeps the chain-rule factors from getting lost in a long expression.

Practice Calculus AB

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.

40 practice questionsEvery one with a written explanation FRQ trainerWritten answers scored against the rubric Calculus AB study guideAll units, weights and score data in one place All 8 Calculus AB unitsThe full course

Questions about this unit

How much of the AP Calculus AB exam is Unit 3?

Unit 3, Composite & Implicit Differentiation, is worth 9–13% of the Calculus AB multiple-choice section according to the published course framework. Across all 8 units that makes it a substantial share — heavier than an even split would give it.

What topics are covered in Calculus AB Unit 3?

Composite & Implicit Differentiation covers Implicit, Inverse functions, Higher-order and Related contexts. We publish 8 terms with definitions for this unit, all of them on this page.

How should I study Calculus AB Unit 3?

Read the 4 lessons below first — about 55 minutes — then drill the 8 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.

All 8 units of AP Calculus AB

Unit names, topics and exam weights follow the published College Board course framework for AP Calculus AB. AP® is a trademark registered by the College Board, which does not endorse this site.