Section 3.1 Derivatives of Polynomials and Exponential Functions

1. Section 3.1Derivatives of Polynomials and Exponential Functions • Goals • Learn formulas for the derivatives of • Constant functions • Power functions • Exponential functions • Learn to find new derivatives from old: • Constant multiples • Sums and differences

2. Constant Functions • The graph of the constant function f(x) = c is the horizontal line y = c … • which has slope 0 , • so we must have f (x) = 0 (see the next slide). • A formal proof is easy:

3. Constant Functions (cont’d)

4. Power Functions • Next we look at the functions f(x) = xn , where n is a positive integer. • If n = 1 , then the graph of f(x) = x is the line y = x , which has slope 1 , so f (x) = 1. • We have already seen the cases n = 2 and n = 3 :

5. Power Functions (cont’d) • For n = 4 we find the derivative off(x) = x4 as follows:

6. Power Functions (cont’d) • There seems to be a pattern emerging! • It appears that in general, if f(x) = xn , then f (x) = nxn -1 . • This turns out to be the case:

7. Power Functions (cont’d) • We illustrate the Power Rule using a variety of notations: • It turns out that the Power Rule is valid for any real number n , not just positive integers:

8. Power Functions (cont’d)

9. Constant Multiples • The following formula says that the derivative of a constant times a function is the constant times the derivative of the function:

10. Sums and Differences • These next rules say that the derivative of a sum (difference) of functions is the sum (difference) of the derivatives:

11. Example

12. Exponential Functions • If we try to use the definition of derivative to find the derivative of f(x) = ax , we get: • The factor ax doesn’t depend on x , so we can take it in front of the limit:

13. Exponential (cont’d) • Notice that the limit is the value of the derivative of f at 0 , that is,

14. Exponential (cont’d) • This shows that… • if the exponential function f(x) = ax is differentiable at 0 , • then it is differentiable everywhere and f (x) = f (0)ax • Thus, the rate of change of any exponential function is proportional to the function itself.

15. Exponential (cont’d) • The table shown gives numerical evidence for the existence of f (0) when • a = 2 ; here apparently f (0) ≈ 0.69 • a = 3 ; here apparently f (0) ≈ 1.10

16. Exponential (cont’d) • So there should be a number a between 2 and 3 for which f (0) = 1 , that is, • But the number e introduced in Section 1.5 was chosen to have just this property! • This leads to the following definition:

17. Exponential (cont’d) • Geometrically, this means that • of all the exponential functions y = ax , • the function f(x) = ex is the one whose tangent at (0, 1) has a slope f (0) that is exactly 1 . • This is shown on the next slide:

18. Exponential (cont’d)

19. Exponential (cont’d) • This leads to the following differentiation formula: • Thus, the exponential function f(x) = ex is its own derivative.

20. Example • If f(x) = ex – x , find f(x) and f (0) . • Solution The Difference Rule gives • Therefore

21. Solution (cont’d) • Note that ex is positive for all x , sof (x) > 0 for all x . • Thus, the graph off is concave up. • This is confirmedby the graph shown.

22. Review • Derivative formulas for polynomial and exponential functions • Sum and Difference Rules • The natural exponential function ex