Short on time? This page is the recap. For explanations, the flow diagram and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- A composite is a function of a function. The inner function is what you would calculate first; the outer is what you do last.
- Chain rule: if h(x) = f(g(x)), then h′(x) = f′(g(x)) · g′(x).
- Three moves: differentiate the outside, leave the inside alone, multiply by the derivative of the inside.
- Leibniz form: dy/dx = (dy/du) · (du/dx). Rates multiply; units cancel like fractions.
- Several layers: work outside in, one factor per layer.
- With products or quotients, decide the last operation first; that sets the outer structure.
- Hidden composites are everywhere: sin²x, e^(4x), ln(3x) and 1/(x² + 1) all need the chain rule. Ask “is the input exactly x?” If not, there is an inner function.
Key relationships
| Function | Derivative |
|---|---|
| [g(x)]ⁿ | n[g(x)]ⁿ⁻¹ · g′(x) |
| √g(x) | g′(x)/(2√g(x)) |
| sin(g(x)) | cos(g(x)) · g′(x) |
| cos(g(x)) | −sin(g(x)) · g′(x) |
| tan(g(x)) | sec²(g(x)) · g′(x) |
| e^(g(x)) | e^(g(x)) · g′(x) |
| ln(g(x)), g(x) > 0 | g′(x)/g(x) |
| f(g(x)) at x = a, from a table | f′(g(a)) · g′(a): find g(a) first |
Assumptions behind the rule
- g must be differentiable at x, and f must be differentiable at the value g(x).
- f′ is evaluated at the output of g, not at x.
- Angles are in radians; the trig derivatives above rely on it.
Mistakes to avoid
- Forgetting g′(x). d/dx [sin(5x)] = 5 cos(5x).
- Changing the inside. d/dx [cos(x²)] = −2x sin(x²), not −sin(2x).
- Using f′(a) instead of f′(g(a)) with a table.
- Confusing sin²x with sin(x²). Their derivatives are 2 sin x cos x and 2x cos(x²).
- Squaring the derivative instead of differentiating the square: d/dx [g(x)]² = 2g(x)g′(x), not [g′(x)]² or 2g′(x).
- Using the quotient rule on 1/(stuff) when (stuff)⁻¹ with the chain rule is faster.
Quick self-check
- Differentiate (4x − 1)⁵. (20(4x − 1)⁴)
- Differentiate sin(x³). (3x² cos(x³))
- Find the derivative of e^(5x) at x = 0. (5)
- In Leibniz form, a rate in metres per second times a rate in kelvin per metre gives what units? (kelvin per second)
- If g(2) = 7, g′(2) = 3 and f′(7) = −1, find the derivative of f(g(x)) at x = 2. (f′(7) · g′(2) = −3)
Next: practice questions.