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Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

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  1. 3.1 The Chain Rule

    1. FUN-3.C—Calculate derivatives of compositions of differentiable functions

      • FUN-3.C Calculate derivatives of compositions of differentiable functions. • FUN-3.C.1 The chain rule provides a way to differentiate composite functions. • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

  2. 3.2 Implicit Differentiation

    1. FUN-3.D—Calculate derivatives of implicitly defined functions

      • FUN-3.D Calculate derivatives of implicitly defined functions. • FUN-3.D.1 The chain rule is the basis for implicit differentiation. • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

  3. 3.3 Differentiating Inverse Functions

    1. FUN-3.E—Calculate derivatives of inverse and inverse trigonometric functions

      • FUN-3.E Calculate derivatives of inverse and inverse trigonometric functions. • FUN-3.E.1 The chain rule and definition of an inverse function can be used to find the derivative of an inverse function, provided the derivative exists. • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

  4. 3.4 Differentiating Inverse Trigonometric Functions

    1. FUN-3.E—Calculate derivatives of inverse and inverse trigonometric functions—Topic 3.4

      • FUN-3.E Calculate derivatives of inverse and inverse trigonometric functions. • FUN-3.E.2 The chain rule applied with the definition of an inverse function, or the formula for the derivative of an inverse function, can be used to find the derivatives of inverse trigonometric functions. • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

  5. 3.5 Selecting Procedures for Calculating Derivatives

    1. 3.5

      • 3.5 This topic is intended to focus on the skill of selecting an appropriate procedure for calculating derivatives. Students should be given opportunities to practice when and how to apply all learning objectives relating to calculating derivatives.

  6. 3.6 Calculating Higher-Order Derivatives

    1. FUN-3.F—Determine higher order derivatives of a function

      • FUN-3.F Determine higher order derivatives of a function. • FUN-3.F.1 Differentiating f′ produces the second derivative f″, provided the derivative of f′ exists; repeating this process produces higher-order derivatives of f. • FUN-3.F.2 Higher-order derivatives are represented with a variety of notations. For y = f(x), notations for the second derivative include d²y/dx², f″(x), and y″. Higher-order derivatives can be denoted dⁿy/dxⁿ or f⁽ⁿ⁾(x). • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.