3.1 Definition of Derivative

1. 3.1 Definition of Derivative Goal: Use definition of derivative to find slope, rate of change, instantaneous velocity at a point.

2. Definition: Thederivative of f(x) with respect to x is the function  and is defined as, This is read “f prime of x”

3. Find the derivative: Hint: use (x+h) and (x) into definition

4. 2. Use definition of derivative to find the slope, rate of change, and velocity: Using the last equation for the derivative, find the slope of the tangent at x=2 Rate of change at x= -3 Instantaneous velocity at x= 9

5. Find the derivative by using the definition of derivative:

6. Find the derivative by rationalization:

7. Find the derivative by finding left and right limits

8. Differentiability • Definition A function is called differentiable at f(x) at x=a if exists and is called differentiable on an interval if the derivative exists for each point in the interval. Theorem If f(x) is differentiable at x=a, then f(x) is continuous at x=a.

9. Different notations referring to the derivative of f(x) with respect to x

10. 6. Determine the graph of the derivative

11. Tangent Line and Normal Line Slope of the tangent line is The normal line is perpendicular to the tangent line. The slope is the opposite reciprocal of the tangent line.

12. Find the tangent line and normal line at x= -1 check graphs 3.1

13. Homework 3.1 InterActMath.com