# Derivatives

**1) Average Rate of Change**

Explanation:

Video: what is a rate?

Interactive websites for exploring the relationship of the average rate of change (secant lines) and the instantaneous rate of change (tangent lines):

Teacherlink (flash)

University of British Columbia - Math Labs: Secant Lines (Java - don't open in Chrome) - move the red point on top of the blue point to find the slope at

*x*= 0

Practice

**2) Definition of a Derivative **

Explanation:

Videos:

Kahn: Calculus: Derivatives 1 (new HD version) - general definition

Kahn: Calculus: Derivatives 2.5 (new HD version) - find the derivative using the definition

Hollis: The derivative (slides #1-5)

Interactive Websites:

http://www.univie.ac.at/future.media/moe/galerie/diff1/diff1.html#ableitung (look at the first applet to look at tangent line only; do not open in Chrome) - move the slider to see how the slope of the tangent line changes

Tangent to a point on a curve (Shodor - Java); use the slider to change the point of tangency after you type in a function to a graph; the slope of the change line changes therefore the slope is a function of x.

Khan Module: video explanation | Derivative Intuition: Tangent to a point on a curve (intuitive approach) and all of the slopes of the tangent lines creates a function

Practice Problems:

**3) Power Rule (and sum/difference, and constant multiple rules)**

Explanation:

Khan video: Calculus: Derivatives 3: Determining the derivatives of simple polynomials.

Practice:

Kahn - Power Rule (integer exponent)

**6) Product and Quotient Rule with Symbolic Notation**

Explanation:

Practice Problems

**9) Chain Rule (advanced)**

Explanation

Practice Problems

**10) Derivative of Exponential & Logarithmic Expressions**

Explanation:

Kahn videos:

Practice Problems:

Khan Practice:

Special Derivatives (includes power rule, ln(x), e^x and trig ratios)

**13) Review Summary: **

**The Chain Rule.**The last operator is what you keep; all of the other operators are "u."When should you use the chain rule:

When you have a power (numerical exponent) and the base is anything other than a single variable.

When you have a trig ratio and the "inside" part is anything other than a single variable

When you have an exponential function (the base is a number) and the exponent is anything other than a single variable.

When you have a logarithmic function and the "inside" part is anything other than a single variable.

**14) Derivative at a point with a calculator **

**15) Implicit differentiation - one dy/dx**

Explanation:

Practice Problems:

**17) Review Implicit Differentiation**

Practice Problems

2004 Q4 - parts a and b

**18) Derivative of Inverse Functions**

Explanation:

Practice Problems:

2007 Q3 - parts a and d

**19) Derivative of Inverse Functions with the graphing calculators**

Explanation

**20) Derivative of Inverse Trig Ratios**

Explanation:

Practice Problems:

**21) Points of Non-Differentiability**

Explanation:

Video: Hollis: The derivative (slide #9)

Practice Problems:

**23) ****Derivative Cumulative**