How do roots and reciprocals use the power rule?
You will be able to: Rewrite powers carefully and retain restrictions on the derivative domain.
BC foundation: Unit 2 shares its derivative foundations with AB. Connect each rule to rates, graphs and its domain conditions. Use the written challenges to explain your reasoning. Chain-rule compositions, implicit and inverse differentiation, and advanced applications come later.
How do roots and reciprocals use the power rule?
A reciprocal decreases as its positive input grows, while a square root still increases but becomes flatter. Negative and fractional exponents express both patterns.
A useful starting point: Why does the power rule lower an exponent? →
Words and symbols before equations
- Negative exponent
- x⁻ⁿ=1/xⁿ, requiring x≠0.
- Fractional exponent
- For example x^(1/2)=sqrt(x) on the real domain x≥0.
- Derivative domain
- Inputs where a finite derivative exists.
- Real cube root
- The real number whose cube is x, including negative x.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Controls use positive inputs to compare all three examples. Text states full domains: reciprocal excludes 0; sqrt derivative requires x>0; cube-root derivative excludes 0 but includes negative x.
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 positive a=1, f(a)=1 and f′(a)=-2. Original and derivative exclude x=0.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Rewrite 1/x² as x⁻². Its derivative is −2x⁻³=−2/x³ for x≠0. The coefficient is negative and the exponent becomes more negative.
For sqrt(x)=x^(1/2), the rule gives (1/2)x^(−1/2)=1/(2sqrt(x)) for x>0. The function includes zero but its finite derivative does not.
For cube root of x, the derivative is 1/[3(cube root of x)²] for x≠0, on both positive and negative inputs. At zero the quotient is unbounded.
Always compare the original domain with the derivative formula’s restrictions. Fractional powers cannot be blindly evaluated at every real number.
A worked example, step by step
Differentiate f(x)=1/x² and g(x)=sqrt(x), then evaluate both derivatives at 4.
- Write f=x⁻² and g=x^(1/2).
- Apply the power rule: f′=−2x⁻³ and g′=(1/2)x^(−1/2).
- At 4: f′(4)=−2/64=−1/32; g′(4)=1/4.
- State domains: f′ for x≠0 and g′ for x>0.
An input can belong to the function’s domain while failing to belong to the finite derivative’s domain.
Is sqrt(x) defined and differentiable at zero in the ordinary finite sense?
Compare with an explanation
It is defined and right-continuous there, but its right quotient is unbounded, so it has no finite derivative there.
Predict. Change one thing. Explain.
Select a reciprocal, square root or cube root. Compare original heights and derivative slopes at valid positive inputs; then use the explanation to identify excluded derivative inputs.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At positive a=1, f(a)=1 and f′(a)=-2. Original and derivative exclude x=0.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Controls use positive inputs to compare all three examples. Text states full domains: reciprocal excludes 0; sqrt derivative requires x>0; cube-root derivative excludes 0 but includes negative x.
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 questionFind the derivative of the real cube root function at −8 and explain what happens at zero.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: For x≠0, derivative=1/[3(cube root of x)²].
- 1 point: Cube root of −8 is −2.
- 1 point: At −8 the derivative is 1/(3·4)=1/12.
- 1 point: At zero the difference quotient grows without bound; no finite derivative exists.
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 x⁻² mean?
1/x² for nonzero x.
RECALL 2How does 1/2 change under the power rule?
It becomes a coefficient and the new exponent is −1/2.
RECALL 3Why state domains?
Algebraic rules do not create missing function values or finite slopes.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do roots and reciprocals use the power rule?
- d(x⁻²)/dx=−2/x³, x≠0.
- d(sqrt(x))/dx=1/(2sqrt(x)), x>0.
Remember: An input can belong to the function’s domain while failing to belong to the finite derivative’s domain.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Controls use positive inputs to compare all three examples. Text states full domains: reciprocal excludes 0; sqrt derivative requires x>0; cube-root derivative excludes 0 but includes negative x.
Refresh Kid · AP Calculus BC Unit 2 · Objectives FUN-3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.5, 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 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.
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