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Section 3.1 Derivatives of Polynomials and Exponential Functions

Section 3.1 Derivatives 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. Constant Functions.

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Section 3.1 Derivatives of Polynomials and Exponential Functions

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  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

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