AP Calculus AB/BC Unit 3 Cheat Sheet: Composite, Implicit & Inverse Functions

The basics

What it covers: The chain rule, implicit differentiation, derivatives of inverse and inverse trig functions, and higher-order derivatives.

Exam weight: About 9–13% of the AB exam and 4–7% of the BC exam.

The big question: How do we differentiate functions the basic rules can't reach directly?

Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.

Key topics at a glance

The Chain Rule

d/dx[f(g(x))] = f′(g(x))·g′(x). Differentiate outside-in: outer derivative (keep the inside) times the inside's derivative.

Chain Rule Shortcuts

sin(u) → cos(u)·u′; e^u → e^u·u′; ln(u) → u′/u; uⁿ → n·uⁿ⁻¹·u′.

Implicit Differentiation

Differentiate both sides w.r.t. x, attaching dy/dx to every y-term (chain rule), then solve for dy/dx. Ex: x²+y²=25 → dy/dx = −x/y.

Inverse Function Derivative

(f⁻¹)′(x) = 1 / f′(f⁻¹(x)). The slope of an inverse is the reciprocal of the original's slope at the matching point.

Inverse Trig Derivatives

arcsin x → 1/√(1−x²); arccos x → −1/√(1−x²); arctan x → 1/(1+x²).

Selecting Procedures

Identify the structure first: composite → chain; product → product rule; quotient → quotient rule; y not isolated → implicit. Messy functions combine several.

Higher-Order Derivatives

Differentiate repeatedly. f″ = (f′)′. Notation: f″, y″, d²y/dx². The 2nd derivative governs acceleration and concavity.

Key Idea

Almost everything here is the chain rule in disguise — implicit differentiation and inverse derivatives both come from it.

The key terms and rules you must know

Chain Rule — d/dx[f(g(x))] = f′(g(x))·g′(x).
Composite function — f(g(x)), a function inside a function.
Implicit differentiation — differentiate both sides w.r.t. x, attach dy/dx to y-terms.
d/dx[yⁿ] — n·yⁿ⁻¹·(dy/dx) implicitly.
Inverse derivative — (f⁻¹)′(x) = 1/f′(f⁻¹(x)).
arcsin x, arccos x — 1/√(1−x²) and −1/√(1−x²).
arctan x — 1/(1+x²).
Higher-order derivative — the derivative of a derivative (f″, f‴, …).
Second derivative — f″ = d²y/dx²; gives acceleration and concavity.
Selecting procedures — choosing chain/product/quotient/implicit for a given function.

Key themes to remember

The chain rule is the engine. Composite, implicit, and inverse derivatives all rely on it.
Attach dy/dx to every y. That single habit makes implicit differentiation routine.
Inverses use reciprocals. Reflecting over y = x swaps rise and run.
Read structure before differentiating. Identify the outermost operation first.
Differentiate repeatedly for higher orders — each derivative is the rate of change of the last.

Common exam traps

Don't forget the inner derivative. d/dx[sin(3x)] = 3cos(3x), not cos(3x).
Every y-term needs dy/dx. Skipping it is the #1 implicit-differentiation error.
Product/quotient terms need their rules even implicitly. d/dx[xy] = y + x(dy/dx).
arccos and arcsin differ by a sign. arccos gets the negative.
Inverse derivative is not 1/f′(x). It is 1/f′ evaluated at f⁻¹(x).
Simplify the inside first if it helps, but keep the chain-rule factor.
Second derivative is the derivative of f′, not f′ squared.