# Category Archives: Derivatives

## Derivative Rules

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Learning Objective 2.1C: Calculate Derivatives

Essential Knowledge 2.1C2: Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.

Essential Knowledge 2.1C3: Sums, differences, products, and quotients of functions can be differentiated using derivative rules.

Mathematical Practice for AP Calculus (MPAC) 1: Reasoning with definitions and theorems

Students can: develop conjectures based on exploration with technology.

How do you provide students the opportunity to develop conjectures?

After determining the derivative of a few quadratic functions using the definition, we use our TI-Nspire Computer Algebra System (CAS) software to explore derivatives.

We use the power rule to make conjectures about the product rule. (I think that I saw this suggestion in a Mathematics Teacher magazine in the early 90s, but I can’t find the reference now.)

We know what the derivative should be, because we know the derivative of x^5. How could we use f, f ‘, g, and g ‘ to get to what we know is the derivative from the power rule?

Once students made conjectures about the product rule, we formalized the rule.

I asked students to predict the derivative of f(x)=sin(3x). As expected, many thought that it would be f ’(x)=cos(3x). When we looked at the graph of the derivative of f(x), students realized that f ‘(x)=3cos(3x).

We used CAS to explore the chain rule (power and composite) in more detail.

Students practiced “Notice and Note”. Several generalized the chain power rule before I asked.

Once students knew the chain rule, we used the chain rule to derive the quotient rule.

And so the journey providing opportunities for students to make sense of rules instead of just telling them rules continues …

Posted by on September 14, 2016 in Calculus, Derivatives

## Seeing the Definition of Derivative

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

1. relate the concept of a limit to all aspects of calculus
2. Students can connect concepts to their visual representation with and without technology.

How do you introduce the definition of a derivative?

We start with the visual of a tangent line at a point and a secant line containing the point.

We don’t need calculus to determine the slope of the secant line. We do need calculus to determine the slope of the tangent line.

How might we use the slope of the secant line to determine the slope of the tangent line?

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Posted by on September 9, 2016 in Calculus, Derivatives

## Seeing the Derivative

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

1. Students can connect concepts to their visual representation with and without technology.

How do you introduce the concept of a derivative?

We start with visuals of the derivative (from a Getting Started with Calculus activity called Derivative Trace).

Tangent Line Demonstration from Calculus Nspired has some similar ideas.

We then look at the Derivative Grapher to connect the slope of the tangent line to the graph of the derivative, changing the original function as requested by the students.

We begin to develop some common language around derivatives before we formalize what is a derivative, before we formalize the definition of a derivative.