1. The Second Derivative Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College 2008 Photo by Vickie Kelly, 2003 Arches National Park

2. Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative.is the fourth derivative. We will learn later what these higher order derivatives are used for. 

3. First derivative: is positive Curve is rising. is negative Curve is falling. is zero Possible local maximum or minimum. Second derivative: is positive Curve is concave up. is negative Curve is concave down. is zero Possible inflection point (where concavity changes).

4. Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College 2008 Differentiability

5. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity

6. Most of the functions we study in calculus will be differentiable.

7. Two theorems: If f has a derivative at x = a, then f is continuous at x = a. Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

8. Intermediate Value Theorem for Derivatives Between a and b, must take on every value between and. If a and b are any two points in an interval on which f is differentiable, then takes on every value between and. 