1. 2.2 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

2. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative of

3. “f prime x”or “the derivative of f with respect to x” “y prime” “dee why dee ecks” or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” or “the derivative of f of x”

4. dx does not mean d times x ! dy does not mean d times y ! does not mean times !

5. does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

6. The derivative is the slope function of the original function. The derivative is defined at the end points of a function on a closed interval. Given

7. GIVEN Find a formula for y’

8. Find the equation of the line tangent to at x = 4

9. A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. 

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

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