What does a derivative of a derivative measure?
You will be able to: Compute and interpret higher derivatives with correct notation and units.
BC foundation: Unit 3 shares these differentiation objectives with AB. Track the input at each stage of a composition, preserve branch and domain conditions, and explain why your chosen derivative rule applies. Parametric and polar derivatives come later.
What does a derivative of a derivative measure?
A cart has position s(t)=t³ meters. At t=2 seconds its velocity is 12 m/s. We can also ask how fast that velocity is changing at that moment.
A useful starting point: How do you combine several derivative rules without losing a term? →
Words and symbols before equations
- Second derivative
- The derivative of the first derivative.
- Higher-order derivative
- A derivative obtained by repeated differentiation.
- f″
- Read f double prime, the second derivative, not the square of f′.
- d²s/dt²
- Leibniz notation for a second derivative with respect to time.
What this picture assumes
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. s=t³ meters for t≥0. Position, velocity and acceleration have different units. Separate graph panels avoid treating these as interchangeable quantities.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- At 2 s: position=8 m, velocity=12 m/s, acceleration=12 m/s². Third derivative=6 m/s³.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Differentiate s=t³ once to obtain s′=3t², the velocity in m/s. Differentiate again to obtain s″=6t, the acceleration in m/s².
At t=2, s=8 m, s′=12 m/s and s″=12 m/s². The equal numerical rate values have different meanings and units.
A third derivative is s‴=6 m/s³, and the fourth is zero for this ideal model. Superscript parentheses f^(n) denote derivative order, not an exponent.
A second derivative exists only where the first derivative is differentiable. The ability to take one derivative does not guarantee that all higher derivatives exist.
| Feature | f′ | f″ |
|---|---|---|
| Meaning | Rate of change of f | Rate of change of f′ |
| For position s(t) | Velocity | Acceleration |
| Units for meters and seconds | m/s | m/s² |
A worked example, step by step
For f(x)=x⁴−3x², find f′, f″ and f‴ at x=1.
- First differentiate: f′=4x³−6x.
- Differentiate again: f″=12x²−6.
- A third differentiation gives f‴=24x.
- At x=1 the values are −2, 6 and 24 respectively.
f″ is not (f′)². Derivative order counts repeated differentiation, not repeated multiplication.
Is acceleration the square of velocity?
Compare with an explanation
No. It is the rate of change of velocity; its units are m/s² rather than m²/s².
Predict. Change one thing. Explain.
Move t for s=t³. Compare position, velocity and acceleration readouts, and explain why equal numerical values need not represent the same quantity.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At 2 s: position=8 m, velocity=12 m/s, acceleration=12 m/s². Third derivative=6 m/s³.
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. s=t³ meters for t≥0. Position, velocity and acceleration have different units. Separate graph panels avoid treating these as interchangeable quantities.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using a difference quotient, tangent slope, or derivative rule with its domain conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFor s(t)=2t³−t²+4 meters, find velocity, acceleration and acceleration at t=2 s.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Velocity is s′=6t²−2t m/s.
- 1 point: Acceleration is s″=12t−2 m/s².
- 1 point: At t=2, acceleration is 22 m/s².
- 1 point: This is a change in velocity per second, not a position or a squared velocity.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does f^(3) mean?
The third derivative of f.
RECALL 2How do you find a second derivative?
Differentiate the first derivative as a new function.
RECALL 3Does one derivative guarantee all higher ones?
No; each next derivative needs its own differentiability.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What does a derivative of a derivative measure?
- f″(x)=d[f′(x)]/dx.
- If f has output units U and input units V, f″ has units U/V².
Remember: f″ is not (f′)². Derivative order counts repeated differentiation, not repeated multiplication.
Conditions: Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. s=t³ meters for t≥0. Position, velocity and acceleration have different units. Separate graph panels avoid treating these as interchangeable quantities.
Refresh Kid · AP Calculus BC Unit 3 · Objectives FUN-3.F · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.6, FUN-3.F. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 3 has six official topics. Topic 3.5 integrates existing derivative objectives and Mathematical Practice 1.C; it does not introduce a separate new objective. Focused lesson titles and questions are original Refresh Kid teaching material.
Chain rule factors are evaluated at their correct nested inputs. Implicit derivatives refer to local branches and retain denominator conditions. Inverse derivatives require the corresponding preimage and a nonzero original derivative under the local inverse conditions. Trigonometric inputs and returned angles use radians; inverse branch conventions and domains are stated. Related rates and L’Hôpital’s rule remain in later units. Higher derivatives are repeated differentiation, not powers.
The Organic Chemistry Tutor video creator and relevant descriptions were checked (the implicit video link is labeled descriptively); full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.6, 3.7 and 3.8 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model folds a graph 180 degrees around y=x to construct its inverse reflection. Intermediate depth is a geometric construction, not an extra function variable. Camera rotation only changes the view. Equal-scale labeled 2D graphs and text provide the complete explanation.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
Want to work through this with a tutor?
Bring your question about What does a derivative of a derivative measure? Your explanation and answers remain free to access.