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.
Fiveable's AP Calculus AB/BC Unit 3 review covers 6 topics, with guided practice questions for the unit.
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.
Try AP-style questions on unit 3, then use the review notes below for anything you miss.
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 |
| 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. |
Miss rates come from Fiveable multiple-choice practice.
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.
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.
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.