Unit 3 Differentiation: Composite, Implicit, and Inverse Functions

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ENDURING UNDERSTANDING

FUN-3 Recognizing opportunities to apply derivative rules can simplify differentiation.

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Topic Name

Essential Knowledge

3.1 The Chain Rule

LEARNING OBJECTIVE

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

FUN-3.C.1 The chain rule provides a way to differentiate composite functions.

Blog Posts

Foreshadowing the Chain Rule and Power Rule Implies Chain Rule

Experimenting with CAS – Chain Rule (7-3-2013) Discovering the Chain Rule

The Chain Rule and another The Chain Rule

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3.2 Implicit Differentiation

LEARNING OBJECTIVE

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

 

FUN-3.D.1 The chain rule is the basis for implicit differentiation.

Blog Posts

Implicit Differentiation  Approaches to teaching about implicit differentiation

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3.3 Differentiating Inverse Functions

LEARNING OBJECTIVE

FUN-3.D Calculate derivatives of implicitly defined functions

FUN-3.D.1 The chain rule is the basis for implicit differentiation.

Blog Posts

The Calculus of Inverses (11-12-2012) Derivatives of the Inverse Trigonometry functions

Implicit Differentiation (9-28-2017) Where to start this topic.

Inverses Graphically and Numerically (11-14-2012) Derivatives of inverses – the hard way and the easy way.

Implicit Differentiation of Parametric Equations (5-17-2014) BC topic.

A Vector’s Derivatives  BC topic.

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3.4 Differentiating Inverse Trigonometric Functions

LEARNING OBJECTIVE

 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.

 Blog Posts

The Calculus of Inverses (11-12-2012) Derivatives of the Inverse Trigonometry functions

Implicit Differentiation (9-28-2017) Where to start this topic.

Inverses Graphically and Numerically (11-14-2012) Derivatives of inverses – the hard way and the easy way.

Implicit Differentiation of Parametric Equations (5-17-2014) BC topic.

A Vector’s Derivatives  BC topic.

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3.5 Selecting Procedures for Calculating Derivatives

LEARNING OBJECTIVE

None

Blog Posts

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3.6 Calculating Higher-Order Derivatives

LEARNING OBJECTIVE

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\left( x \right) ,notations for  the second derivative include \frac{{{{d}^{2}}y}}{{d{{x}^{2}}}},{{f}'}'\left( x \right), , and  {{y}'}' . Higher-order derivatives can be denoted \frac{{{{d}^{n}}y}}{{d{{x}^{n}}}} or {{f}^{{\left( n \right)}}}\left( x \right)

 Blog Posts