Definition of the Derivative

1. Definition of the Derivative Section 3.1a

2. Answers to the “Do Now” – Quick Review, p.101 1. 7. 2. 8. 3. 9. No, the one-sided limits at x = 1 are different 4. 5. Slope: 10. No, f is discontinuous at x = 1 because 6. DNE

3. Find a formula for the slope of the tangent at any point (a, 1/a) on the graph of the function f(x) = 1/x : This study of the rates of change of functions is called differential calculus, and the formula was our first look at a derivative…

4. Definition: Derivative The derivative of the function with respect to the variable x is the function whose value at x is provided the limit exists. Note: The domain of may be smaller than that of . If exists, we say that has a derivative (is differentiable) at x. A function that is differentiable at every point of its domain is a differentiable function.

5. Applying the Definition Differentiate (that is, find the derivative of)

6. A reminder of how we found the derivative graphically Slope of the secant line PQ:

7. Now, let’s see what happens when we re-label our diagram Slope of the secant line PQ: And what limit should we take here to find the derivative???

8. Definition (Alternate): Derivative at a Point The derivative of the function f at the point x = ais the limit provided the limit exists. If we use this alternate form to find the derivative of f at x = a, we can find the general derivative of f by applying our answer to an arbitrary x in the domain of f…………………observe……

9. Applying Both Definitions Differentiate using both derivative definitions. The original definition:

10. Applying Both Definitions Differentiate using both derivative definitions. The alternate definition: Applying this formula to an arbitrary x > 0 in the domain of f identifies the derivative as the function

11. Notes about Notation Besides , the most common notations for the derivative of a function : Notation Reads Notes “y prime” Nice and brief, but does not name the independent variable. “dy/dx” or “the derivative of y with respect to x” Names both variables and uses d for derivative.

12. Notes about Notation Besides , the most common notations for the derivative of a function : Notation Reads Notes “df/dx” or “the derivative of f with respect to x” Emphasizes the function’s name. “d dxof f at x” or “the derivative of f at x” Emphasizes the idea that differentiation is an operation performed on f.

13. One-Sided Derivatives A function y = f(x) is differentiable on a closed interval [a,b] if it has a derivative at every interior point of the interval, and if the limits [the right-hand derivative at a] [the left-hand derivative at b] exist at the endpoints. Note: In the right-hand derivative, h is positive and a + h approaches a from the right. In the left-hand derivative, h is negative and b + h approaches b from the left.

14. One-Sided Derivatives Derivatives at endpoints are one-sided limits: Slope = Slope =

15. One-Sided Derivatives Show that the following function has left-hand and right-hand derivatives at x= 0, but no derivative there. Graph the function: The left-hand derivative: The right-hand derivative: Since these are not equal, the function has no derivative at x = 0!!!