Definition of the Derivative

1. Definition of the Derivative Lesson 3.4

2. Tangent Line • Recall from geometry • Tangent is a line that touches the circle at only one point • Let us generalize the concept to functions • A tangent will just "touch" the line but not pass through it • Which of the above lines are tangent?

3. A Secant Line • Crosses the curve twice • The slope of the secant will be x = a h

4. Tangent Line • Now let h get smaller and smaller x = a Try this with the TI Nspire again h The slope of the tangent line

5. Tangent Line • Try it out …given • Determine the slope of the tangent line at x = 0 • Evaluate

6. Tangent Line • Use the limit command on your calculator to determine the slope offor x = 1 • Once you have the slope and the point • You can determine the equation of the line

7. The Derivative • We will define the derivative of f(x) as • Note • The derivative is the rate of change function for f(x) • The derivative is also a function of x • The limit must exist

8. Comparison Difference Quotient • Slope of secant • Average rate of change • Average velocity Derivative • Slope of tangent • Instantaneous rate of change • Instantaneous velocity

9. Finding f'(x) from Definition • Strategy • Find f(x + h) • Find and simplify f(x + h) – f(x) • Divide by h to get • Let h → 0

10. Try It Out • Use the strategy to find the derivatives of these functions Hint: rationalize the numerator

11. Equation of the Tangent Line • We stated previously that once we determine the slope of the tangent • We can "cut to the chase" and state it as

12. Warning • Our definition of derivative includedthe phrase "if the limit exists" • Derivatives do not exist at "corners" or "sharp points" on the graph • The slope is different on each side of the point • The limit does not exist f(x) = | x – 3 |

13. Assignment • Lesson 3.4 • Page 210 • Exercises 1 – 51 odd