Differentiation: Composite, Implicit, and Inverse Functions

Unit Overview

This unit is dominated by one enormously powerful idea: the Chain Rule, which differentiates composite functions (a function inside another). Once you have it, you can differentiate almost anything, and you can extend implicit differentiation and inverse-function differentiation to a much wider set of problems — including the inverse trigonometric functions.

The Chain Rule is arguably the most tested single rule in all of AP Calculus. Expect it woven through nearly every subsequent unit (related rates, optimization, integration by substitution).

Big picture: To differentiate f(g(x)), differentiate the outside function (leaving the inside alone) and multiply by the derivative of the inside.

Learning Objectives

• Apply the Chain Rule to composite functions, including nested compositions.
• Combine the Chain Rule with the Product and Quotient Rules.
• Extend implicit differentiation using the Chain Rule.
• Differentiate inverse functions and the inverse trigonometric functions.
• Select and combine the correct procedure(s) for any derivative.
• Compute higher-order derivatives of composite and implicit functions.

Prerequisite Concepts

• All fundamental rules from Unit 2 (power, product, quotient, trig).
• Function composition; recognizing "inside" and "outside" functions.
• The inverse-function relationship and basic inverse-trig values.

Key Definitions

• Composite function: (f ∘ g)(x) = f(g(x)) — g is the inside, f the outside.
• Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x). In Leibniz form: dy/dx = (dy/du)(du/dx).
• Implicit function: y defined by an equation not solved for y (e.g., x² + y² = 25).
• Inverse trig functions: arcsin, arccos, arctan, the inverses of the (restricted) trig functions.

Important Theorems / Rules

Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x).

Chain Rule with a table: If h(x) = f(g(x)), then h'(a) = f'(g(a)) · g'(a).

Inverse Function Derivative: (f⁻¹)'(a) = 1 / f'(f⁻¹(a)).

Important Theorems / Rules

Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x).

Chain Rule with a table: If h(x) = f(g(x)), then h'(a) = f'(g(a)) · g'(a).

Inverse Function Derivative: (f⁻¹)'(a) = 1 / f'(f⁻¹(a)).

Important Formulas

Chain Rule versions of the basics (u = u(x)):

• d/dx[u^n] = n·u^(n−1)·u'
• d/dx[sin u] = cos u · u' (similarly for the other trig functions)
• d/dx[e^u] = e^u · u'
• d/dx[ln u] = u'/u

Inverse trigonometric derivatives:

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

With an inside function u: multiply each by u' (Chain Rule), e.g. d/dx[arctan(u)] = u'/(1 + u²).

Concept Explanations

Peeling the onion (3.1). For a composite, identify layers from the outside in. Differentiate the outermost layer (keeping its inside intact), then multiply by the derivative of what's inside — repeating for each layer. Example: sin(cos x) → cos(cos x) · (−sin x).

Chain Rule powers implicit differentiation (3.2). Every time you differentiate a term containing y with respect to x, the Chain Rule attaches a dy/dx. That is the whole mechanism: d/dx[y³] = 3y²·(dy/dx).

Inverse functions without inverting (3.3). Reflecting a graph over y = x swaps rise and run, so slopes are reciprocals at corresponding points — which is exactly (f⁻¹)'(a) = 1/f'(f⁻¹(a)).

Inverse trig derivatives (3.4). These are algebraic even though the functions are transcendental — memorize the three AB forms. Note arccos is just the negative of arcsin's derivative.

Selecting procedures (3.5). Complex expressions often need several rules at once: e.g., cos(3x)/eˣ needs the Quotient Rule and the Chain Rule on the numerator. Plan the outer structure first (product? quotient? composite?), then handle each piece.

Worked Examples

Example 1 (Basic Chain Rule). d/dx[(3x + 1)⁵] = 5(3x + 1)⁴ · 3 = 15(3x + 1)⁴.

Example 2 (Trig composite). d/dx[cos(x²)] = −sin(x²) · 2x = −2x sin(x²).

Example 3 (Nested). d/dx[sin(cos x)] = cos(cos x) · (−sin x) = −cos(cos x) sin x.

Example 4 (Table). If g(2) = 3, g'(2) = 4, f'(3) = 5, and h = f∘g, then h'(2) = f'(g(2))·g'(2) = 5·4 = 20.

Example 5 (Implicit). For x² + xy + y² = 3, differentiate: 2x + (y + x·y') + 2y·y' = 0. At (1, 1): 2 + 1 + y' + 2y' = 0 → 3 + 3y' = 0 → y' = −1.

Example 6 (Inverse trig + product). d/dx[x·arctan x] = arctan x + x/(1 + x²).

Example 7 (Inverse function). For f(x) = x⁵ + 2x + 1 with inverse g, find g'(4). Since f(1) = 4, g(4) = 1. f'(x) = 5x⁴ + 2, so f'(1) = 7. Thus g'(4) = 1/7.

Graphical Interpretation

• A composite's tangent slope is the product of the layer slopes.
• Inverse functions are reflections over y = x; corresponding tangent slopes are reciprocals (a steep curve has a shallow inverse and vice versa).
• Where f'(f⁻¹(a)) = 0 (a horizontal tangent of f), the inverse has a vertical tangent (its derivative blows up).

Common Mistakes

• Forgetting the · u' factor — the single most common Chain Rule error (writing d/dx[sin(2x)] = cos(2x) instead of 2cos(2x)).
• Not attaching dy/dx to y-terms in implicit differentiation.
• Sign error on arccos — its derivative is negative.
• Inverse-derivative mix-up: g'(a) = 1/f'(g(a)), evaluated at the inverse's input, not at a directly.
• Stopping too early on nested functions — differentiate every layer.

AP Exam Tips

• Say the Chain Rule to yourself as "derivative of the outside, times the derivative of the inside."
• Table problems (given values of f, g, f', g') are almost always Chain Rule or product/quotient — read which composition is asked.
• For inverse-function values, first find the matching input, then reciprocate f' there.
• Memorize arcsin, arccos, arctan derivatives; they show up in both differentiation and (reverse) as integrals in Unit 6.

Unit Summary

• Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x) — differentiate outside, times derivative of inside.
• It powers implicit differentiation (every y-term gets a dy/dx) and extends to inverse and inverse-trig derivatives.
• Know the three inverse-trig derivatives and combine rules deliberately for complex expressions.
• (f⁻¹)'(a) = 1/f'(f⁻¹(a)).

Next unit: apply derivatives to real contexts — motion, related rates, linearization, and L'Hôpital's Rule.