How do you differentiate a whole polynomial?
You will be able to: Apply constant, sum, difference and constant-multiple rules term by term.
How do you differentiate a whole polynomial?
A position rule s(t)=2t³−3t+5 combines a curved contribution, a linear contribution and a starting offset. Each part contributes differently to its rate.
A useful starting point: How do roots and reciprocals use the power rule? →
Words and symbols before equations
- Constant multiple
- A fixed number times a function.
- Sum rule
- Derivative of a sum equals the sum of the derivatives.
- Difference rule
- Derivative of a difference equals the difference of the derivatives.
- Vertical shift
- Adding a constant to every output; it does not change slopes.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f=Ax³+Bx+C; f′=3Ax²+B. All coefficients are constants with respect to x. The constant offset changes heights but not slopes.
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.
- f=(1)x³+(-1)x+(1); f′=(3)x²+(-1). At a=1, height=1 and slope=2. C changes only heights; A and B change slopes.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Limits distribute over sums and fixed multiples, so differentiation does too. If f and g are differentiable, (af+bg)′=af′+bg′ for constants a and b.
Differentiate 2t³−3t+5 term by term: 2·3t²−3·1+0=6t²−3.
The constant offset changes position heights but cancels from every output difference, so it contributes zero to velocity.
These linear rules do not say (fg)′=f′g′. Products of changing factors require a separate rule in Topic 2.8.
A worked example, step by step
Find p′(x) and p′(−1) for p(x)=4x³−2x²+7.
- Differentiate 4x³ to get 12x².
- Differentiate −2x² to get −4x; the constant 7 gives 0.
- Combine p′(x)=12x²−4x.
- At −1: 12+4=16.
A constant multiplier stays; an added constant vanishes. Keep subtraction signs attached to their terms.
Do x²+100 and x² have the same derivative?
Compare with an explanation
Yes. Their constant separation contributes no output change, so both derivatives are 2x.
Predict. Change one thing. Explain.
Change the cubic coefficient, linear coefficient and constant offset separately. Predict which control moves heights without changing slopes.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
f=(1)x³+(-1)x+(1); f′=(3)x²+(-1). At a=1, height=1 and slope=2. C changes only heights; A and B change slopes.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f=Ax³+Bx+C; f′=3Ax²+B. All coefficients are constants with respect to x. The constant offset changes heights but not slopes.
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 questionDifferentiate q(x)=−2x⁴+3x²−6x+9 and evaluate q′(1).
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The quartic derivative is −8x³.
- 1 point: The quadratic and linear derivatives are 6x and −6.
- 1 point: The constant gives 0, so q′=−8x³+6x−6.
- 1 point: At 1, q′(1)=−8+6−6=−8.
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 1Derivative of c f(x) for constant c?
c f′(x).
RECALL 2Derivative of f−g?
f′−g′.
RECALL 3Do these rules apply to a product of changing functions?
No. Use the product rule.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you differentiate a whole polynomial?
- (af+bg)′=af′+bg′ for constants a,b.
- For p=ax³+bx+c, p′=3ax²+b.
Remember: A constant multiplier stays; an added constant vanishes. Keep subtraction signs attached to their terms.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f=Ax³+Bx+C; f′=3Ax²+B. All coefficients are constants with respect to x. The constant offset changes heights but not slopes.
Refresh Kid · AP Calculus AB Unit 2 · Objectives FUN-3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.6, FUN-3.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 2 has 10 official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative definitions require finite two-sided limits at interior inputs. Graphs and finite tables supply evidence and estimates, not universal proofs. Continuity is necessary but insufficient for differentiability. Rules retain their original domain restrictions, and trigonometric inputs use radians. Chain-rule compositions, implicit differentiation, inverse-function differentiation and L’Hôpital’s rule are deferred to later units. Piecewise joins are checked for continuity before their side slopes are compared.
The Organic Chemistry Tutor video titles, creators and descriptions were checked; 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 3.1, 3.3 and 3.5 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 uses separate planes for f and f′. Depth distinguishes panels, not an additional variable; camera rotation does not change their values. Use the complete labeled 2D graphs and text to read precise coordinates and slopes.
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 AB 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.
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