Differential calculus: techniques and applications
Twelve lessons extending calculus foundations through derivative rules, motion, theorem-based analysis and optimization. This is a differential calculus unit, not complete Calculus I or AP AB/BC preparation. Written arguments require rubric-based review.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
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Differential calculus: mixed methods assessment
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence
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Trigonometric derivatives need radian measure
Lesson details and objectives
Connect circle geometry and limiting rates; preserve domain and angle units.
- Differentiate sine, cosine and the other basic trigonometric functions.
- Explain why the standard formulas assume radians.
- Use quotient identities and diagnose degree-mode errors.
6 practice tasks.
Exponential and logarithmic rates
Lesson details and objectives
Connect multiplicative growth, inverse functions and logarithmic differentiation.
- Differentiate exponentials and natural logarithms with domains.
- Distinguish a base from an exponent and use change of base.
- Use logarithmic differentiation for a variable exponent.
6 practice tasks.
Products and quotients change in two ways
Lesson details and objectives
Derive and apply rules that account for changes in both factors.
- Derive the product rule from a difference quotient.
- Apply the quotient rule with nonzero denominator.
- Diagnose missing terms and reversed signs.
6 practice tasks.
Follow a change through nested functions
Lesson details and objectives
Track every intermediate rate and avoid treating derivative notation as unrestricted cancellation.
- Differentiate nested functions with their domains.
- Explain chain factors with units.
- Distinguish a valid chain-rule argument from a missing-factor shortcut.
6 practice tasks.
Differentiate a relationship without solving it
Lesson details and objectives
Track dependent variables and distinguish horizontal, vertical and singular points.
- Differentiate an implicit equation using the chain rule.
- Use the equation to check that a point lies on the curve.
- Recognize when a formula for dy/dx is unavailable.
6 practice tasks.
Rates for inverse functions and inverse trigonometry
Lesson details and objectives
Use a local inverse carefully, with nonzero slope and selected branches.
- Derive the inverse derivative relation.
- Differentiate arcsine, arccosine and arctangent on their valid domains.
- Distinguish inverse functions from reciprocals.
6 practice tasks.
Position, velocity, acceleration and higher rates
Lesson details and objectives
Use signs and units to distinguish moving, speeding up and changing direction.
- Compute higher derivatives with units.
- Analyze velocity and acceleration signs.
- Distinguish zero velocity, rest over an interval and reversal.
6 practice tasks.
Relate simultaneous rates before substituting
Lesson details and objectives
Build a geometric relationship, differentiate in time, then evaluate an instant.
- Identify changing quantities and fixed parameters.
- Differentiate a constraint before substituting instantaneous values.
- Interpret signs, units and model validity.
6 practice tasks.
Rolle and the mean value theorem
Lesson details and objectives
Turn endpoint information into a justified intermediate-rate guarantee.
- Check closed-interval continuity and open-interval differentiability.
- Apply Rolle and the mean value theorem without claiming uniqueness.
- Use a derivative bound to bound a change.
6 practice tasks.
Read increase and curvature from derivative signs
Lesson details and objectives
Combine sign charts with domain information rather than relying on a plotted appearance.
- Use first derivatives for increase and extrema.
- Use second derivatives for concavity and inflection.
- Handle inconclusive second-derivative tests and missing domain points.
6 practice tasks.
Compare all candidates before choosing a design
Lesson details and objectives
Use existence hypotheses, boundaries and contextual constraints in optimization.
- Apply the extreme value theorem and closed-interval candidate method.
- Build an objective and feasible domain from a scenario.
- Distinguish stationary, boundary, unattained and global optima.
6 practice tasks.
Indeterminate forms and justified derivative ratios
Lesson details and objectives
Check the form and theorem hypotheses before replacing a ratio by a ratio of derivatives.
- Use L’Hopital for eligible 0/0 and infinity/infinity limits.
- Rewrite other indeterminate forms before applying the theorem.
- Recognize invalid applications and inconclusive derivative-ratio limits.
6 practice tasks.