Calculating Higher-Order Derivatives
Differentiate rates again while preserving chain rules, implicit dependence, notation, and units.
College Board Topic 3.6 · CED p. 75 (PDF p. 80) · CED effective Fall 2020; project course outline 2026–27.
A derivative can itself change. Differentiate f′ to obtain f″ wherever that derivative exists, and repeat for f‴ or higher derivatives. For y=f(x), common notations are y″, f″(x), d²y/dx², and f⁽ⁿ⁾(x) for the nth derivative. The superscript order in f⁽ⁿ⁾ is not an ordinary power.
For f(x)=x⁴, f′=4x³, f″=12x², f‴=24x and f⁽⁴⁾=24. For sine, derivatives cycle: sin x, cos x, −sin x, −cos x, sin x. Each stage needs the same rule discipline as a first derivative; a chain-rule factor appears again when an inside function is differentiated again.
In a position model s(t), s′ is velocity and s″ is acceleration; units change from meters to meters/second to meters/second². When y is defined implicitly, differentiate the first-derivative equation while continuing to treat y as a function of x. Higher derivatives are meaningful only where the previous derivative exists and is differentiable.
Let's work through it
- For f(x)=eˣsin x, apply the product rule to get f′=eˣ(sin x+cos x). Reason: both factors vary.
- Differentiate again: f″=eˣ(sin x+cos x)+eˣ(cos x−sin x). Reason: the first derivative remains a product.
- Combine to get f″=2eˣcos x. Reason: opposite sine terms cancel.
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
For f(x)=x³, what is f″(x)?
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
For f(x)=x³, what is f″(x)?
Worked answer and reason
6x.
Differentiate 3x² a second time.
Retrieval 2
Does f⁽⁴⁾(x) mean [f(x)]⁴?
Worked answer and reason
No. It means the fourth derivative.
Derivative-order notation uses parentheses to distinguish it from a power.
Catch the tempting mistake
- The second derivative is the square of the first derivative.
- It is the derivative of the first derivative; it describes how that rate changes.
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
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+ − * / ^ ! · 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 27 · No calculator · analytic
- Unit 3 problem 28 · No calculator · verbal
- Unit 3 problem 29 · No calculator · analytic
- Unit 3 problem 30 · No calculator · analytic
For the teacher
- Use units and derivative-order notation in contextual examples.
- Include a second implicit derivative so students retain the chain rule beyond the first pass.
Glossary and deeper reading · 4 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.
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Velocity in calculus
In one-dimensional motion, velocity is the derivative of position with respect to time. Its sign records direction relative to the chosen positive axis. For example, s(t)=t²−4t meters gives velocity 2t−4 meters per second. Speed is the magnitude of velocity and therefore cannot be negative.
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Acceleration in calculus
Acceleration is the derivative of velocity with respect to time. In one-dimensional motion, it is also the second derivative of position. For example, s(t)=t² meters gives constant acceleration 2 meters per second squared. A negative acceleration can increase speed when the velocity is also negative.