44 To find the derivative of the function we can write, f (x) = x 2 f (x) = 2x f (x) = x 3 f (x) = 3x 2 This pattern generalizes to any power of x: Theorem 4.1 The Power Rule If n is any constant and f (x) = x n, then f (x) = nx n – 1. Quick Example If f (x) = x 2, then f (x) = 2x 1 = 2x.
55 Example 1 – Using the Power Rule for Negative and Fractional Exponents Calculate the derivatives of the following: Solution: a. Rewrite as f (x) = x –1. Then f (x) = (–1)x –2 b. Rewrite as f (x) = x –2. Then f (x) = (–2)x –3
66 c. Rewrite as f (x) = x 0.5. Then f (x) = 0.5x –0.5 Alternatively, rewrite f (x) as x 1/2, so that cont’d Example 1 – Solution
77 By rewriting the given functions in Example 1 before taking derivatives, we converted them from rational or radical form (as in, say, and ) to exponent form (as in x –2 and x 0.5.) Shortcut Formula: The Power Rule
99 Differential notation (“d-notation”) is based on an abbreviation for the phrase “the derivative with respect to x.” For example, we learned that if f (x) = x 3, then f (x) = 3x 2. When we say “f (x) = 3x 2,” we mean the following: The derivative of x 3 with respect to x equals 3x 2.
10 Differential Notation; Differentiation means “the derivative with respect to x.” Thus, [f (x)] is the same thing as f (x), the derivative of f (x) with respect to x. If y is a function of x, then the derivative of y with respect to x is (y) or, more compactly, To differentiate a function f (x) with respect to x means to take its derivative with respect to x. Another Notation: Differential Notation
11 Quick Example In Words Formula The derivative with respect (x 3 ) = 3x 2 to x of x 3 is 3x 2. Another Notation: Differential Notation
13 The Rules for Sums and Constant Multiples We can now find the derivatives of more complicated functions, such as polynomials, using the following rules: Theorem 4.2 Derivatives of Sums, Differences, and Constant Multiples If f (x) and g(x) are any two differentiable functions, and if c is any constant, then the functions f (x) + g(x) and cf (x) are differentiable, and [f (x) ± g(x)] = f (x) ± g (x) [cf (x)] = cf (x). Sum Rule Constant Multiple Rule
14 In Words: The derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives. The derivative of c times a function is c times the derivative of the function. Differential Notation: The Rules for Sums and Constant Multiples
15 Quick Example The Rules for Sums and Constant Multiples
17 By the definition of the derivative of a function, Proof of the Sum Rule
18 The next-to-last step uses a property of limits: the limit of a sum is the sum of the limits. Think about why this should be true. The last step uses the definition of the derivative again (and the fact that the functions are differentiable). The proof of the rule for constant multiples is similar. Proof of the Sum Rule
19 Example 2 – Combining the Sum and Constant Multiple Rules, and Dealing with x in the Denominator Find the derivatives of the following: a. f (x) = 3x 2 + 2x – 4 b. Solution: a. Rule for sums Rule for differences
20 b. Notice that f has x and powers of x in the denominator. By rewriting them in exponent form (that is, in the form constant power of x): We are now ready to take the derivative: cont’d Rational form Exponent form Example 2 – Solution
21 Example 2 – Solution Rational form cont’d Exponent form
22 The Derivative of a Constant Times x and the Derivative of a Constant If c is any constant, then: Rule Quick Example Proof of the Sum Rule
23 Derivative of | x | Note that the derivative does not exist when x = 0. Quick Example Proof of the Sum Rule
25 Example 4 – Gold Price You are a commodities trader and you monitor the price of gold on the New York Spot Market very closely during an active morning. Suppose you find that the price of an ounce of gold can be approximated by the function G(t) = –8t 2 + 144t + 150 dollars (7.5 ≤ t ≤ 10.5) where t is time in hours.
27 a. According to the model, how fast was the price of gold changing at 10:00 AM ? b. According to the model, the price of gold (A) increased at a faster and faster rate (B) increased at a slower and slower rate (C) decreased at a faster and faster rate (D) decreased at a slower and slower rate between 9:30 and 10:30 AM. cont’d Example 4 – Gold Price
28 Example 4(a) – Solution Differentiating the given function with respect to t gives G (t) = –16t + 144. Because 10:00 AM corresponds to t = 10, we obtain G (10) = –16(10) + 144 = –16. The units of the derivative are dollars per hour, so we conclude that, at 10:00 AM, the price of gold was dropping at a rate of $16 per hour.
29 Example 4(b) – Solution From the graph, we can see that, between 9:30 and 10:30 AM (the interval [9.5, 10.5]),the price of gold was decreasing. Also from the graph, we see that the slope of the tangent becomes more and more negative as t increases, so the price of gold is decreasing at a faster and faster rate (choice (C)). cont’d
30 Example 4(b) – Solution We can also see this algebraically from the derivative, G (t) = –16t + 144: For values of t larger than 9, G (t) is negative; that is, the rate of change of G is negative, so the price of gold is decreasing. Further, as t increases, G (t) becomes more and more negative, so the price of gold is decreasing at a faster and faster rate, confirming that choice (C) is the correct one. cont’d
32 An Application to Limits: L’Hospital’s Rule The limits are those of the form lim x→a f (x) in which substituting x = a gave us an indeterminate form, such as L’Hospital’s rule gives us an alternate way of computing limits such as these without the need to do any preliminary simplification. It also allows us to compute some limits for which algebraic simplification does not work. Substituting x = 2 yields Substituting x = yields
33 An Application to Limits: L’Hospital’s Rule Theorem 4.3 L’Hospital’s Rule If f and g are two differentiable functions such that substituting x = a in the expression gives the indeterminate form then That is, we can replace f (x) and g(x) with their derivatives and try again to take the limit.
34 An Application to Limits: L’Hospital’s Rule Quick Example Substituting x = 2 in yields Therefore, l’Hospital’s rule applies and
35 Example 5 – Applying L’Hospital’s Rule Check whether l’Hospital’s rule applies to each of the following limits. If it does, use it to evaluate the limit. Otherwise, use some other method to evaluate the limit.
36 Setting x = 1 yields Therefore, l’Hospital’s rule applies and We are left with a closed-form function. However, we cannot substitute x = 1 to find the limit because the function (2x – 2)/(12x 2 – 6x – 6) is still not defined at x = 1. Example 5(a) – Solution
37 Example 5(a) – Solution In fact, if we set x = 1, we again get 0/0. Thus, l’Hospital’s rule applies again, and Once again we have a closed-form function, but this time it is defined when x = 1, giving Thus cont’d
38 Example 5(b) – Solution Setting x = yields so Setting x = again yields so we can apply the rule again to obtain Note that we cannot apply l’Hospital’s rule a third time because setting x = yields the determinate form 4/ = 0. Thus, the limit is 0. cont’d
39 Example 5(c) – Solution Setting x = 1 yields 0/0 so, by l’Hospital’s rule, We are left with a closed-form function that is still not defined at x = 1. Further, l’Hospital’s rule no longer applies because putting x = 1 yields the determinate form 1/0. cont’d
40 Example 5(d) – Solution Setting x = 1 in the expression yields the determinate form 1/0, so l’Hospital’s rule does not apply here. Using the methods of limits and continuity, we find that the limit does not exist. cont’d