The Chain Rule
Identify composition layers, evaluate each rate at the correct input, and include every inside factor.
College Board Topic 3.1 · CED p. 70 (PDF p. 75) · CED effective Fall 2020; project course outline 2026–27.
A composition changes in layers. If y=f(u) and u=g(x), then dy/dx=(dy/du)(du/dx), provided the component derivatives exist at the relevant inputs. In function notation, (f∘g)′(x)=f′(g(x))g′(x). Differentiate the outside while leaving the inside in place, then multiply by the inside rate.
For y=(3x²+1)⁴, the outside rule gives 4(3x²+1)³ and the inside rate is 6x, so y′=24x(3x²+1)³. For y=sin(x²), y′=cos(x²)·2x. For y=e^(−x), y′=−e^(−x). Missing the inside factor is the most common error.
More layers mean more factors. For y=ln(1+e^(2x)), the layers are log, addition, exponential and 2x. The derivative is [1/(1+e^(2x))]·e^(2x)·2. With table data, evaluate the outside derivative at the inside output: if g(2)=5, the outside derivative needed is f′(5), not f′(2). Units can reveal the structure: meters per liter times liters per minute gives meters per minute.
Let's work through it
- For y=sin(3x²), identify outside sin u and inside u=3x². Reason: the expression is a composition.
- The outside derivative is cos(3x²); the inside derivative is 6x. Reason: differentiate each layer at its own input.
- Multiply to get y′=6x cos(3x²). Reason: the chain rule combines the rates through the layers.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Differentiate (2x+1)⁵.
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Differentiate (2x+1)⁵.
Worked answer and reason
10(2x+1)⁴.
The outside factor is 5(2x+1)⁴ and the inside rate is 2.
Retrieval 2
If g(1)=4, g′(1)=−2 and f′(4)=3, find (f∘g)′(1).
Worked answer and reason
−6.
Evaluate the outside derivative at g(1)=4, then multiply by g′(1).
Catch the tempting mistake
- Differentiate the outside and ignore the inside because it is unchanged.
- The inside remains in the outside formula, but its rate still multiplies the result.
Make the picture move
Explore → predict → explain
A derivative is a limit of slopes.
Try first. At the selected a, predict the tangent slope. Which side’s secant lies above it, and which lies below?
Scroll sideways to see the full graph on a small screen. The table below gives the numerical values.
- Function: x²
- Selected secant
- Exact tangent
Right secant slope 2.5 at a = 1. The derivative is 2.
Compare the numbers
| Distance |h| | Left secant slope | Right secant slope |
|---|---|---|
| 0.5 | 1.5 | 2.5 |
| 0.25 | 1.75 | 2.25 |
| 0.1 | 1.9 | 2.1 |
| 0.05 | 1.95 | 2.05 |
What’s going on? For x², the difference quotient simplifies to 2a + h. As h approaches zero from either side, the secant slope approaches 2a.
Keep in mind. The controls keep h nonzero so that the quotient is defined. A small h is an approximation; the limiting calculation supplies the exact derivative.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 3 problem 1 · No calculator · analytic
- Unit 3 problem 2 · No calculator · analytic
- Unit 3 problem 3 · No calculator · analytic
- Unit 3 problem 4 · No calculator · analytic
- Unit 3 problem 5 · No calculator · numerical
- Unit 3 problem 6 · No calculator · multiple
- Unit 3 problem 7 · Graphing calculator required · multiple
For the teacher
- Use a labeled layer diagram before symbolic calculation.
- Mix analytic formulas with table and unit-based chain reasoning.
Glossary and deeper reading · 2 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.