Review AP Calculus AB/BC Unit 3 to build fluency with the chain rule and its applications to composite, implicit, and inverse functions. This unit is the backbone of all differentiation work that follows, from related rates to integration techniques.
Use the topic guides, practice questions, and FRQ practice available on Fiveable to work through every differentiation procedure in this unit.
Unit 3 picks up where Unit 2 left off by introducing the chain rule, the single most important differentiation technique in AP Calculus. Once you have the chain rule, implicit differentiation and inverse function derivatives follow directly from it.
This unit teaches you how to differentiate composite functions using the chain rule, how to differentiate equations where y is not isolated using implicit differentiation, how to differentiate inverse and inverse trig functions, how to choose the right rule for any derivative problem, and how to compute second and higher-order derivatives.
Every major technique in this unit relies on the chain rule. Implicit differentiation applies it to y-terms. Inverse function derivatives use it to derive the reciprocal slope formula. Inverse trig derivatives are derived by combining it with trig identities. If your chain rule is shaky, address that first.
Differentiate both sides with respect to x, attach dy/dx to every y-term via the chain rule, collect all dy/dx terms on one side, and solve algebraically. The equation does not need to be solved for y first. Common targets include circles, ellipses, and cubic curves like x^3 + y^3 = 6xy.
A second derivative f''(x) is just the derivative of f'(x). You apply the same power, product, quotient, and chain rules again to whatever you got the first time. Second derivatives connect to concavity and acceleration in Units 4 and 5.
The chain rule formula d/dx[f(g(x))] = f'(g(x)) times g'(x), or in Leibniz notation dy/dx = dy/du times du/dx, is the organizing idea of the entire unit. Recognizing composite structure in a function is the skill that unlocks implicit differentiation, inverse derivatives, inverse trig derivatives, and higher-order work. Every topic in Unit 3 is a different context for applying that same rule.
Differentiate composite functions using d/dx[f(g(x))] = f'(g(x)) times g'(x). Identify the outer and inner functions, differentiate outside-in, and multiply by the derivative of the inner function. Leibniz notation dy/dx = (dy/du)(du/dx) is also tested.
Differentiate both sides of an equation with respect to x, attach dy/dx to every y-term via the chain rule, then solve algebraically for dy/dx. Used for curves like x^2 + y^2 = r^2 and x^3 + y^3 = 6xy. Tangent and normal line problems are common applications.
Use (f^(-1))'(x) = 1 / f'(f^(-1)(x)). The slope of an inverse at a point is the reciprocal of the original function's slope at the corresponding point. Table problems that give f(a) and f'(a) and ask for (f^(-1))'(b) are a frequent exam format.
Know the six inverse trig derivative formulas. The most tested are d/dx[arcsin x] = 1/sqrt(1-x^2), d/dx[arccos x] = -1/sqrt(1-x^2), and d/dx[arctan x] = 1/(1+x^2). Apply the chain rule when the argument is a composite expression.
Read the function's structure before differentiating. Identify whether the outermost operation is a product, quotient, composition, or implicit relationship, then apply rules in the correct order. Logarithmic differentiation handles variable-exponent cases like y = x^x.
Differentiate repeatedly using the same rules. The second derivative f''(x) is the derivative of f'(x). Notation includes f''(x), y'', d^2y/dx^2, and f^(n)(x). Second derivatives connect to concavity in Unit 5 and acceleration in Unit 4.
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Review Implicit Differentiation with attention to how the concept appears in AP-style source and evidence questions.
Review Differentiating Inverse Functions with attention to how the concept appears in AP-style source and evidence questions.
Review Selecting Procedures for Calculating Derivatives with attention to how the concept appears in AP-style source and evidence questions.
Review The Chain Rule with attention to how the concept appears in AP-style source and evidence questions.
A composite function has the form f(g(x)), one function inside another. The chain rule says: differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function. In Leibniz notation, if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). The outside-in strategy means you never change the inside until after you have differentiated the outside.
Can you differentiate d/dx[e^(x^2)] and d/dx[ln(2x+1)] without looking at notes? Both require recognizing the composite structure before applying the chain rule.
| Function | Outer f | Inner g | Derivative |
|---|---|---|---|
| (3x+1)^5 | u^5 | 3x+1 | 15(3x+1)^4 |
| sin(3x) | sin(u) | 3x | 3cos(3x) |
| e^(x^2) | e^u | x^2 | 2x e^(x^2) |
| ln(2x+1) | ln(u) | 2x+1 | 2/(2x+1) |
When an equation like x^2 + y^2 = 1 defines y implicitly rather than explicitly, you differentiate both sides with respect to x. Every y-term picks up a dy/dx factor from the chain rule because y is treated as a function of x. After differentiating, collect dy/dx terms on one side and solve algebraically. Do not substitute a specific point until after you have the general dy/dx expression.
Differentiate x^2 + y^2 = 25 implicitly and find the slope of the tangent line at (3, 4). Then try x^3 + y^3 = 6xy.
| Term | d/dx expression | Key step |
|---|---|---|
| y^2 | 2y dy/dx | Chain rule on y |
| sin y | cos y dy/dx | Chain rule on y |
| e^y | e^y dy/dx | Chain rule on y |
| xy (product) | y + x dy/dx | Product rule, then chain rule on y |
If f and f^(-1) are inverses, the derivative of f^(-1) at x equals 1 divided by f' evaluated at f^(-1)(x). Geometrically, the tangent slopes of f and f^(-1) at corresponding points are reciprocals because the graphs are reflections across y = x. On the AP exam, table problems often give you f(a) and f'(a) and ask for the derivative of f^(-1) at a related point.
Given f(2) = 5 and f'(2) = 3, find (f^(-1))'(5). Then explain why you use f'(2) rather than f'(5).
| Given | What you need | Formula step |
|---|---|---|
| f(a) = b, f'(a) = k | (f^(-1))'(b) | 1/k |
| f(a) = b, f'(a) = 0 | (f^(-1))'(b) | Does not exist |
Each inverse trig function has a specific derivative formula derived by applying the inverse function derivative formula and a Pythagorean identity. The three most tested are arcsin, arccos, and arctan. Note that arccos and arccot derivatives are the negatives of arcsin and arctan derivatives respectively. When the argument is a composite expression, apply the chain rule after the base formula.
Write all six inverse trig derivative formulas from memory. Then differentiate d/dx[arctan(5x)] and d/dx[arcsin(x^2)] using the chain rule.
| Function | Derivative | Sign |
|---|---|---|
| arcsin x | 1/sqrt(1-x^2) | positive |
| arccos x | -1/sqrt(1-x^2) | negative |
| arctan x | 1/(1+x^2) | positive |
| arccot x | -1/(1+x^2) | negative |
| arcsec x | 1/(|x|sqrt(x^2-1)) | positive |
Most AP derivative problems do not label which rule to use. The skill is reading the function's structure first: is it a product, quotient, composition, implicit equation, or inverse? Identify the outermost operation, apply the matching rule, then handle inner layers. Many problems require two or three rules in sequence. Logarithmic differentiation is useful when a variable appears in both the base and the exponent.
For each function, name the rules needed before computing: (a) y = (x^2 + 1)^3 sin x, (b) x^2 y + y^3 = 4, (c) y = arctan(e^x), (d) y = x^x.
| Function type | Primary rule | Secondary rule if needed |
|---|---|---|
| f(x) times g(x) | Product rule | Chain rule on each factor |
| f(x) / g(x) | Quotient rule | Chain rule on numerator or denominator |
| f(g(x)) | Chain rule | Rule matching f or g |
| F(x,y) = 0 | Implicit differentiation | Chain rule on y-terms |
| y = x^x | Logarithmic differentiation | Chain rule after ln |
A higher-order derivative is found by differentiating repeatedly. The second derivative f''(x) is the derivative of f'(x), the third is the derivative of f''(x), and so on. The same differentiation rules apply at every step. Notation varies: f''(x), d^2y/dx^2, and y'' all mean the second derivative. Higher-order derivatives connect to concavity and acceleration in Units 4 and 5.
Find f''(x) for f(x) = x^4 - 3x^2 + 2. Then find d^2y/dx^2 for x^2 + y^2 = 25 using implicit differentiation twice.
| Notation | Meaning | Example |
|---|---|---|
| f'(x) | First derivative | d/dx[x^3] = 3x^2 |
| f''(x) | Second derivative | d/dx[3x^2] = 6x |
| f'''(x) | Third derivative | d/dx[6x] = 6 |
| f^(n)(x) | nth derivative | d^n y/dx^n |
Try AP-style multiple-choice questions and written prompts after you review the notes.
| Term | Definition |
|---|---|
| Composite Function | A function of the form f(g(x)), where one function is applied to the output of another. The chain rule is required to differentiate composite functions. |
| First Derivative | The result of differentiating a function once, representing the instantaneous rate of change or slope of the tangent line at any point. |
| implicitly defined function | A function defined by an equation relating x and y rather than by an explicit formula y = f(x). Implicit differentiation is used to find dy/dx without isolating y. |
| Arctan function | The inverse tangent function, denoted arctan(x) or tan^(-1)(x). Its derivative is d/dx[arctan(x)] = 1/(1 + x^2). |
| arctan | Shorthand for the inverse tangent function. When composed with another function g(x), the chain rule gives d/dx[arctan(g(x))] = g'(x)/(1 + [g(x)]^2). |
| Logarithmic differentiation | A technique where you take the natural log of both sides of y = f(x), differentiate implicitly, and solve for dy/dx. Useful when a variable appears in both the base and the exponent, such as y = x^x. |
| f(x) | Standard function notation where x is the input and f(x) is the output. In Unit 3, f(g(x)) denotes a composite function requiring the chain rule for differentiation. |
A common error is writing d/dx[(3x+1)^5] = 5(3x+1)^4 and stopping. The chain rule requires multiplying by the derivative of the inner function, giving 15(3x+1)^4. Always check whether the inside is more than just x.
When differentiating y^2 with respect to x, the result is 2y dy/dx, not just 2y. Missing the dy/dx factor makes it impossible to solve for the derivative correctly.
To find (f^(-1))'(b), you need f'(a) where f(a) = b, not f'(b). Students often substitute b directly into f' instead of finding the matching input on f first.
The derivatives of arccos, arccot, and arccsc are the negatives of arcsin, arctan, and arcsec respectively. Mixing up the signs is one of the most frequent errors on inverse trig problems.
For a function like (x^2)(sin(3x)), the product rule applies first, then the chain rule handles sin(3x). Applying the chain rule to the whole expression before recognizing the product structure leads to incorrect results.
The chain rule rarely appears alone on the AP exam. Expect it as one step inside a product rule, quotient rule, or implicit differentiation problem. Free-response questions often require you to differentiate a composite expression as part of finding a tangent line, a rate of change, or a related rates setup.
A common multiple-choice and free-response task gives a table of f(x) and f'(x) values and asks for (f^(-1))'(b) at a specific value. The skill tested is correctly identifying the matching input and applying the reciprocal slope formula, not just plugging into a formula blindly.
Higher-order derivatives appear in questions about concavity, inflection points, and acceleration. You may be asked to find f''(x) for an explicitly or implicitly defined function, then interpret its sign. This connects Unit 3 computation directly to the analytical reasoning tasks in Units 4 and 5.
Given f(g(x)), identify the outer and inner functions, differentiate outside-in, and multiply by g'(x). Practice with exponential, trig, logarithmic, and polynomial compositions.
Differentiate both sides with respect to x, attach dy/dx to every y-term, collect dy/dx terms, and solve. Verify by substituting a point back into the original equation before finding slope.
Given a table of f values and f' values, find (f^(-1))'(b) by locating the input a where f(a) = b, then computing 1/f'(a).
Write d/dx[arcsin x], d/dx[arccos x], d/dx[arctan x], d/dx[arccot x], d/dx[arcsec x], and d/dx[arccsc x] from memory. Note the negative signs on co-functions and apply the chain rule when the argument is composite.
For any function, name the outermost structure first. Practice problems that require product plus chain, quotient plus chain, or implicit plus chain in sequence.
Differentiate f'(x) to get f''(x) using the same rules. Practice finding d^2y/dx^2 for both explicit functions and implicit equations.
Know that f''(x) > 0 means concave up and f''(x) < 0 means concave down. In motion contexts, the second derivative of position gives acceleration.
Read the 3.1 topic guide and practice differentiating composite functions involving polynomials, exponentials, trig, and logarithms. Make sure you can identify outer and inner functions quickly before moving on.
Use the 3.2 topic guide to practice the full implicit differentiation process on circles, ellipses, and cubic curves. Find dy/dx and then find the slope of a tangent line at a specific point.
Topics 3.3 and 3.4 share the same underlying formula. Use the topic guides for both, write all six inverse trig formulas from memory, and practice table-based inverse derivative problems.
Work through the 3.5 topic guide using mixed derivative problems. For each function, name the rules needed before computing. Practice problems that combine product, quotient, chain, and implicit differentiation.
Use the 3.6 topic guide to practice finding second and third derivatives for explicit and implicit functions. Then use available practice questions and FRQ practice to test yourself across all six topics under timed conditions.
Open the individual guides for Unit 3 when you want a closer review of one topic.
Use AP-style practice after you review the notes so you can check what you understand.
Practice free-response reasoning and compare your answer with scoring guidance.
Watch past review streams filtered to Unit 3 when you want a video walkthrough.
Open the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
Estimate your broader AP score goal after you review the course and exam format.
AP Calc Unit 3 covers 6 topics: the Chain Rule, Implicit Differentiation, Differentiating Inverse Functions, Differentiating Inverse Trigonometric Functions, Selecting Procedures for Calculating Derivatives, and Calculating Higher-Order Derivatives. The chain rule is the backbone of the unit and connects directly to every other topic here. See the full topic breakdown at AP Calc Unit 3.
Unit 3 makes up 9-13% of the AP Calc exam, making it one of the more heavily tested differentiation units. It covers composite functions via the chain rule, implicit differentiation, inverse and inverse trig derivatives, and higher-order derivatives. Expect to see these concepts woven into both multiple-choice and free-response questions.
The AP Calc Unit 3 progress check in AP Classroom includes both MCQ and FRQ parts drawn from the unit's 6 topics: the chain rule, implicit differentiation, inverse function derivatives, inverse trig derivatives, selecting differentiation procedures, and higher-order derivatives. The MCQ part tests procedural fluency, while the FRQ part asks you to show full differentiation work and justify steps. Practice with matched problems at AP Calc Unit 3.
Unit 3 FRQs most often target implicit differentiation, the chain rule applied to composite functions, and higher-order derivatives. A typical question gives you an equation or a table of values and asks you to find a derivative, justify a result, or find a second derivative. To practice, work through problems that require you to show every differentiation step clearly, since partial credit depends on your written work. Find Unit 3 FRQ practice at AP Calc Unit 3.
For AP Calc Unit 3 practice questions, including multiple-choice and practice test problems, head to AP Calc Unit 3. You'll find MCQ sets covering the chain rule, implicit differentiation, inverse trig derivatives, and higher-order derivatives, plus FRQ-style problems that mirror what shows up on the actual exam.
Start with the chain rule (Topic 3.1) before anything else, since every other topic in the unit builds on it. Once the chain rule clicks, implicit differentiation and inverse function derivatives will feel much more manageable. Work through these steps: (1) practice chain rule problems until the Leibniz notation dy/dx = (dy/du)(du/dx) feels automatic, (2) move to implicit differentiation and remember to apply the chain rule every time you differentiate a y term, (3) memorize the inverse trig derivative formulas and practice recognizing when to use them, (4) finish with higher-order derivatives so you can handle second and third derivative questions. Do a mix of MCQ for speed and FRQ for written justification. Review at AP Calc Unit 3.