AP Calculus AB and BC

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Exam weighting: AB 5-10% · BC 5-10%

AP Calculus Unit 3 teaches you to differentiate functions built from other functions: composites (the chain rule), implicitly defined curves, and inverse and inverse trig functions. It is worth 5-10% of both the AB and BC exams, but the chain rule underlies nearly every derivative you take after this unit.

Every topic in Unit 3 is the chain rule wearing a different costume. Unit 2 gave you rules for plain functions; Unit 3 handles what happens when functions are nested inside each other, tangled together implicitly, or run in reverse. The College Board files all six topics under enduring understanding FUN-3: recognizing which derivative rule applies can turn a scary-looking problem into a routine one.

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}\left[f(g(x))\right] = f'(g(x)) \cdot g'(x)
  1. 3.1 The Chain Rule: differentiate composite functions, the foundation for everything else.
  2. 3.2 Implicit Differentiation: differentiate equations where yy is not isolated.
  3. 3.3 Differentiating Inverse Functions: use the reciprocal-slope relationship g′(x)=1f′(g(x))g'(x) = \frac{1}{f'(g(x))}.
  4. 3.4 Differentiating Inverse Trigonometric Functions: the arcsin⁡\arcsin, arctan⁡\arctan, and arcsec⁡\operatorname{arcsec} formulas.
  5. 3.5 Selecting Procedures for Calculating Derivatives: choose the right rule for any given derivative.
  6. 3.6 Calculating Higher-Order Derivatives: differentiate again to reach f′′(x)f''(x) and beyond.

Topic 3.1 is the engine. A composite function is a function inside a function, like sin⁡(x2)\sin(x^2) or (3x+1)5(3x+1)^5. The recognition cue: if evaluating it by hand means doing one thing, then doing something to the result, it is a composite and needs the chain rule. Differentiate the outer function, leave the inside untouched, then multiply by the derivative of the inside. In Leibniz form, the rule for dydx\frac{dy}{dx} is:

dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Topics 3.2 through 3.4 are all the chain rule in disguise. Implicit differentiation (3.2) applies when yy is tangled into the equation and cannot be solved for cleanly, like x2+y2=25x^2 + y^2 = 25: differentiate both sides with respect to xx, and every yy term produces a dydx\frac{dy}{dx} that you solve for. For inverse functions (3.3), the graph is a reflection, so the slope g′(x)g'(x) is the reciprocal of the original slope at the matching point:

g′(x)=1f′(g(x))g'(x) = \frac{1}{f'(g(x))}

Applying that to the trig functions gives the inverse trig derivatives of 3.4, such as the derivative of arcsin⁡x\arcsin x:

ddxarcsin⁡x=11−x2\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}

Topic 3.5 is where the unit's teaching pays off: given any derivative, name the rule before you compute. Is it a product, a quotient, a composite, or a combination of those? This is a decision skill, not a formula. Topic 3.6 adds higher-order derivatives, differentiating a second time to get f′′(x)f''(x) or d2ydx2\frac{d^2y}{dx^2} (and f(n)(x)f^{(n)}(x) in general), which Unit 5 later uses for concavity and the second derivative test.

What the exam asks

Unit 3 is 5-10% of both the AB and BC exams, but the chain rule shows up far more often than that because it is buried inside Unit 4 related rates, Unit 6 uu-substitution, and any product or quotient with a composite inside. Expect multiple-choice items that ask you to differentiate a nested function quickly, and free-response parts that use implicit differentiation to find a tangent-line slope dydx\frac{dy}{dx} at a point.

Topics in this unit

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

  • 3.1The Chain Rule
  • 3.2Implicit Differentiation
  • 3.3Differentiating Inverse Functions
  • 3.4Differentiating Inverse Trigonometric Functions
  • 3.5Selecting Procedures for Calculating Derivatives
  • 3.6Calculating Higher-Order Derivatives

How to study this unit

  • For Topic 3.1, name the outer and inner function out loud before differentiating: for $\sqrt{4x^2+1}$ the outer is the square root and the inner is $4x^2+1$. Getting that split right is most of the chain rule.
  • In Topic 3.2, remember that differentiating a $y$ term always attaches $\frac{dy}{dx}$ (chain rule) while differentiating an $x$ term does not. After differentiating, collect every $\frac{dy}{dx}$ on one side and factor it out.
  • Do not memorize all six inverse trig derivatives in Topic 3.4 cold. Learn the three ($\arcsin$, $\arctan$, $\operatorname{arcsec}$); the cofunction versions ($\arccos$, $\operatorname{arccot}$, $\operatorname{arccsc}$) are just their negatives. On the AP exam, $\arcsin$, $\arccos$, and $\arctan$ are the ones that actually show up, so prioritize those.
  • Topic 3.5 is a skill, not a formula, so drill mixed sets where you cannot predict the rule. Before computing each one, label it product, quotient, chain, or a combination, then execute.
  • For Topic 3.6, mind your notation: the second derivative is $f''(x)$ or $\frac{d^2y}{dx^2}$. On implicit problems you usually must substitute $\frac{dy}{dx}$ back in before the second derivative simplifies.

Guides for this unit

Tools and tables for this unit