Differentiation tools
What students will learn
Choose derivative rules for explicit, composite, inverse, and implicit functions while respecting domains.
Before this topic
- Derivative definition and notation
- Function composition and inverse relations
Find a resource or check in this book →
Unit 3: Differentiation: Composite, Implicit, and Inverse FunctionsUnit 4: Contextual Applications of Differentiation
Read and connect
Read: Applying the Power Rule →
Apply the power rule to integer, negative, and fractional powers on the valid real domain.
Read: Derivative Rules: Constant, Sum, Difference, and Constant Multiple →
Differentiate constants and linear combinations, preserving signs before point evaluation.
Read: Derivatives of cos x, sin x, e^x, and ln x →
Use familiar trigonometric, exponential, and logarithmic derivatives, including disguised derivative limits.
Read: The Product Rule →
Account for both changing factors in a product, whether values come from formulas or data.
Read: The Quotient Rule →
Choose a quotient rule or equivalent rewrite and preserve the original denominator restrictions.
Read: Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions →
Derive reciprocal-trigonometric rules from identities and track signs, radians, and domains.
Read: The Chain Rule →
Identify composition layers, evaluate each rate at the correct input, and include every inside factor.
Read: Implicit Differentiation →
Differentiate an implicit relation locally, checking the point and any division used to find its slope.
Read: Differentiating Inverse Functions →
Find the corresponding original input before taking a nonzero inverse slope reciprocal.
Read: Differentiating Inverse Trigonometric Functions →
Derive inverse-trigonometric rates on stated branches and distinguish function from derivative domains.
Read: Selecting Procedures for Calculating Derivatives →
Choose and combine rules from the outer structure, rather than treating a mixed expression as one memorized pattern.
Read: Calculating Higher-Order Derivatives →
Differentiate rates again while preserving chain rules, implicit dependence, notation, and units.
Read: Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms →
Check an indeterminate quotient and theorem hypotheses before comparing numerator and denominator derivatives.
Check understanding
Use the complete prompt to make student reasoning visible, then consult the separate worked reasoning and criteria. A reading completion tick records reading history.
Choose rules for a mixed derivative →
Identify and apply the power, product, sine, and chain rules with every changing factor included.
Find an inverse rate from a table →
Choose the original input matching the inverse input before taking the reciprocal slope.
Justify repeated L’Hospital steps →
Identify the indeterminate form and verify the conditions before each numerator-and-denominator differentiation.
Reject an invalid L’Hospital step →
Explain why a one-over-zero form fails the hypothesis and determine the one-sided unbounded behavior.
Checks when the rule is invalid; it does not assess a qualifying indeterminate quotient.
Standards and earlier reading lists
These topic placements organize reading. Edition-qualified Standards mappings and the existing complete unit task sets retain their reviewed scope.