Wemazi

AP Calculus · Unit 2

Differentiation

The definition of the derivative and the basic rules.

Topic 2.1Average versus instantaneous rate

  1. Calculate an average rate from a formula or tableUse endpoint outputs and the full input interval.
  2. Calculate an instantaneous rateEvaluate a derivative at the requested input.
  3. Interpret the sign and unitsExplain increasing or decreasing behavior with output units per input unit.
  4. Compare the two kinds of rateIdentify what each rate describes and when their values match.

Topic 2.2Derivative as a limit

  1. Build a derivative limitForm a correct difference quotient at a point or at a variable input.
  2. Recognize a derivative inside a limitIdentify the function, evaluation point, and any scale factor.
  3. Evaluate a derivative from its definitionSimplify and take the limit for polynomial, radical, or other supported expressions.
  4. Check whether the defining limit existsCompare the relevant one-sided difference quotients when necessary.

Topic 2.3Estimate a derivative

  1. Estimate a derivative from nearby dataDivide the output change by the actual separation of the selected inputs.
  2. Choose a left, right, or centered estimateUse points appropriate to the requested input and side.
  3. Read a tangent slope from a graphTranslate the displayed rise and run into a derivative estimate.
  4. Recognize the limits of an estimateDo not claim an exact derivative from finite data without additional assumptions.

Topic 2.4When differentiability fails

  1. Check continuity before differentiabilityRecognize that a discontinuity prevents a derivative at that point.
  2. Compare one-sided slopesDecide whether a corner or piecewise join has a derivative.
  3. Recognize an unbounded slopeDistinguish a vertical tangent or cusp from a finite derivative.
  4. Make a piecewise join differentiableMatch both the function values and the one-sided derivatives.
  5. Use the correct logical implicationDistinguish differentiability implying continuity from its false converse.

Topic 2.5Power-rule fluency

  1. Differentiate positive integer powersMultiply by the exponent and reduce it by one.
  2. Differentiate negative and fractional powersRewrite reciprocals or radicals when useful and preserve domain restrictions.
  3. Evaluate a power-function slopeSubstitute the requested input into the derivative.
  4. Use a slope to determine an unknownRecover a coefficient or another supported parameter from a derivative condition.

Topic 2.6Differentiate sums and constants

  1. Differentiate a sum or differenceApply differentiation separately to every term.
  2. Handle constants correctlyKeep constant multipliers and differentiate constant terms to zero.
  3. Rewrite powers before differentiatingExpress supported reciprocals and radicals in a useful power form.
  4. Use the derivative at a pointCalculate a slope or determine a coefficient from a slope condition.

Topic 2.7Core trig, exponential and log derivatives

  1. Differentiate sine and cosineUse the correct derivative and the negative sign for cosine.
  2. Differentiate exponential functionsUse the appropriate factor for the exponential base.
  3. Differentiate logarithmic functionsUse the appropriate reciprocal and base factor on the valid domain.
  4. Combine and evaluate these derivativesDifferentiate a linear combination and calculate its rate at a point.

Topic 2.8Differentiate a product

  1. Differentiate a productInclude both product-rule terms with the correct unchanged factors.
  2. Use a product rule with dataCombine function values and derivatives from a table at the same input.
  3. Apply a product derivativeFind a tangent slope or a rate in a changing-quantity problem.
  4. Find a missing value in the ruleUse a known product derivative to recover a missing function value or derivative.

Topic 2.9Differentiate a quotient

  1. Differentiate a quotientUse the correct subtraction order and square the denominator.
  2. Use a quotient rule with dataSubstitute function and derivative values from a table, checking the denominator.
  3. Use a quotient derivative at a pointFind a slope or rate while respecting the original domain.
  4. Check a simplified derivativeRecognize an equivalent expression without restoring excluded inputs.

Topic 2.10Other trigonometric derivatives

  1. Differentiate tangent and secantUse the correct trigonometric derivative rules.
  2. Differentiate cotangent and cosecantKeep the negative signs and the correct products.
  3. Evaluate a trigonometric slopeUse exact trig values when possible and check that the derivative is defined.
  4. Recognize equivalent derivative formsUse identities to compare or simplify answers correctly.