# Cluster 2: Differential Calculus

Develop an understanding of the derivative as an instantaneous rate of change, using geometrical, numerical, and analytical methods. Use this definition to find derivatives of algebraic and transcendental functions and combinations of these functions (using, for example, sums, composites, and inverses). Find second and higher order derivatives. Understand and use the relationship between differentiability and continuity. Understand and apply the Mean Value Theorem. Find derivatives of algebraic, trigonometric, logarithmic, and exponential functions. Find derivatives of sums, products, and quotients, and composite and inverse functions. Find derivatives of higher order, and use logarithmic differentiation and the Mean Value Theorem.
General Information
Number: MAFS.912.C.2
Title: Differential Calculus
Type: Cluster
Subject: Mathematics
Domain-Subdomain: Calculus

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Tutorials

Calculus: Derivatives 1:

In this video we will learn that a derivative is simply the slope of a curve at any given point.

Type: Tutorial

Calculating Slope of Tangent Line Using Derivative Definition:

In this video we will use find the slope of the tangent line to determine the derivative.

Type: Tutorial

Mean Value Theorem:

We will learn and apply the Mean Value Theorem.

Type: Tutorial

Derivative as Slope of a Tangent Line:

We will find the derivative of a function by finding the slope of the tangent line.

Type: Tutorial

Mean Value Theorem:

In this video we will take an in depth look at the Mean Value Theorem.

Type: Tutorial

The Derivative of f(x)=x^2 for Any x:

In this video we will find the derivative of a function based on the slope of the tangent line.

Type: Tutorial

Using the Product Rule and the Chain Rule:

In this video we will use the chain rule and the product rule together to find a derivative of a composite function.

Type: Tutorial

The Product Rule for Derivatives:

In this video will will apply the product rule to find the derivative of two functions.

Type: Tutorial

Product Rule for More Than Two Functions:

In this video, we will use the product rule to find the derivative of the product of three functions.

Type: Tutorial

Derivative of Log with Arbitrary Base:

In this video, we will find the derivative of a log with an arbitrary base.

Type: Tutorial

Chain Rule for Derivative of 2^x:

Here we will see how the chain rule is used to find the derivative of a logarithmic function.

Type: Tutorial

Chain Rule Introduction:

This video is an introduction on how to apply the chain rule to find the derivative of a composite function.

Type: Tutorial

Chain Rule Definition and Example:

In this video we will define the chain rule and use it to find the derivative of a function.

Type: Tutorial

Chain Rule With Triple Composition:

We will use the chain rule to find the derivative of a triple-composite function.

Type: Tutorial

Chain Rule Example Using Visual Information:

In this video we will analyze the graph of a function and its tangent line, then use the chain rule to find the value.

Type: Tutorial

Chain Rule Example Using Visual Function Definitions:

We will use the chain rule to find the value of a composite function at a given point.

Type: Tutorial

## Video/Audio/Animation

MIT BLOSSOMS - The Physics of Boomerangs:

This learning video explores the mysterious physics behind boomerangs and other rapidly spinning objects. Students will get to make and throw their own boomerangs between video segments! A key idea presented is how torque causes the precession of angular momentum. One class period is required to complete this learning video, and the optimal prerequisites are a familiarity with forces, Newton's laws, vectors and time derivatives. Each student would need the following materials for boomerang construction: cardboard (roughly the size of a postcard), ruler, pencil/pen, scissors, protractor, and a stapler.

Type: Video/Audio/Animation

## Virtual Manipulative

Derivative Plotter:

This online applet that depicts the derivative of a given function. Can use demo examples or a user-defined function.

Type: Virtual Manipulative

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Tutorials

Calculus: Derivatives 1:

In this video we will learn that a derivative is simply the slope of a curve at any given point.

Type: Tutorial

Calculating Slope of Tangent Line Using Derivative Definition:

In this video we will use find the slope of the tangent line to determine the derivative.

Type: Tutorial

Mean Value Theorem:

We will learn and apply the Mean Value Theorem.

Type: Tutorial

Derivative as Slope of a Tangent Line:

We will find the derivative of a function by finding the slope of the tangent line.

Type: Tutorial

Mean Value Theorem:

In this video we will take an in depth look at the Mean Value Theorem.

Type: Tutorial

The Derivative of f(x)=x^2 for Any x:

In this video we will find the derivative of a function based on the slope of the tangent line.

Type: Tutorial

Using the Product Rule and the Chain Rule:

In this video we will use the chain rule and the product rule together to find a derivative of a composite function.

Type: Tutorial

The Product Rule for Derivatives:

In this video will will apply the product rule to find the derivative of two functions.

Type: Tutorial

Product Rule for More Than Two Functions:

In this video, we will use the product rule to find the derivative of the product of three functions.

Type: Tutorial

Derivative of Log with Arbitrary Base:

In this video, we will find the derivative of a log with an arbitrary base.

Type: Tutorial

Chain Rule for Derivative of 2^x:

Here we will see how the chain rule is used to find the derivative of a logarithmic function.

Type: Tutorial

Chain Rule Introduction:

This video is an introduction on how to apply the chain rule to find the derivative of a composite function.

Type: Tutorial

Chain Rule Definition and Example:

In this video we will define the chain rule and use it to find the derivative of a function.

Type: Tutorial

Chain Rule With Triple Composition:

We will use the chain rule to find the derivative of a triple-composite function.

Type: Tutorial

Chain Rule Example Using Visual Information:

In this video we will analyze the graph of a function and its tangent line, then use the chain rule to find the value.

Type: Tutorial

Chain Rule Example Using Visual Function Definitions:

We will use the chain rule to find the value of a composite function at a given point.

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.