Derivative of Logarithmic Functions

9 Key excerpts on "Derivative of Logarithmic Functions"

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  eBook - PDF

  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  DERIVATIVES OF LOGARITHMIC FUNCTIONS We will establish that f (x) = ln x is differentiable for x > 0 by applying the derivative def- inition to f (x). To evaluate the resulting limit, we will need the fact that ln x is continuous for x > 0 [Theorem 1.8.2(b)], and we will need the limit lim v →0 (1 + v) 1∕v = e (1) that was given in Formula (6) of Section 1.8 (with x rather than v as the variable). Using the definition of the derivative, we obtain d dx [ln x] = lim h →0 ln (x + h) − ln x h = lim h →0 1 h ln ( x + h x ) The quotient property of logarithms in Theorem 1.8.3 = lim h →0 1 h ln ( 1 + h x ) = lim v →0 1 vx ln (1 + v) Let v = h∕x and note that v →0 if and only if h →0. = 1 x lim v →0 1 v ln (1 + v) x is fixed in this limit computation, so 1∕x can be moved through the limit sign. = 1 x lim v →0 ln (1 + v) 1∕v The power property of logarithms in Theorem 1.8.3 = 1 x ln [ lim v →0 (1 + v) 1∕v ] ln x is continuous on (0, +∞) so we can move the limit through the function symbol. = 1 x ln e = 1 x Since ln e = 1 136 Chapter 3 / Topics in Differentiation Thus, d dx [ln x] = 1 x , x > 0 (2) A derivative formula for the general logarithmic function log b x can be obtained from (2) by using Formula (9) of Section 1.8 to write d dx [log b x] = d dx [ ln x ln b ] = 1 ln b d dx [ln x] It follows from this that d dx [log b x] = 1 x ln b , x > 0 (3) Note that, among all possible bases, the base b = e produces the sim- plest formula for the derivative of log b x. This is one of the reasons why the natural logarithm function is preferred over other logarithms in calculus. 1 2 3 4 5 6 −1 1 x y y = ln x with tangent lines Figure 3.2.1 Example 1 (a) Figure 3.2.1 shows the graph of y = ln x and its tangent lines at the points x = 1 2 , 1, 3, and 5. Find the slopes of those tangent lines. (b) Does the graph of y = ln x have any horizontal tangent lines? Use the derivative of ln x to justify your answer.

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  eBook - ePub

  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  x ), then such a function must be a new function other than a power function. We ask the question:

  Is there any function whose derivative is (1/x )?12

  Note that, we have obtained the function whose derivative is (1/x ). Thus, is the desired (new ) function that fills up the gap noticed above. We call it the natural logarithm function .

  Recall that the definition of logarithmic function was encountered in algebra and it was based on exponents. The properties of logarithms were then proved from the corresponding properties of exponents.13

  Definition: The natural logarithmic function denoted by ln (or loge ) is defined by

  The properties of logarithms can be proved by means of this definition. However, to understand this definition, we have to study the properties of definite integrals and the first fundamental theorem of calculus. These topics are discussed in Part II of this book.

  Now, let us try to differentiate

  Consider,

  Differentiating both the sides with respect to y , we get

  This gives Now, for reverting to the original function, we use the formula We get

  Thus, for we have

  Next, let us try to differentiate First, we must change

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  Applied Calculus

  • Geoffrey Berresford, Andrew Rockett(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  119–120. Use the preceding formulas to find the derivative of each function. The rules for differentiating logarithmic functions with (positive) base a are as follows: For example, d dx log 5 x  1 (ln 5) x d dx log 2 ( x 3  1)  3 x 2 (ln 2)( x 3  1) 119. a. f ( x )  10 x b. f ( x )  3 x 2  1 c. f ( x )  2 3 x d. f ( x )  5 3 x 2 e. f ( x )  2 4  x 120. a. f ( x )  5 x b. f ( x )  2 x 2  1 c. f ( x )  3 4 x d. f ( x )  9 5 x 2 e. f ( x )  10 1  x Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 290 Chapter 4 Exponential and Logarithmic Functions These formulas are more complicated than the corresponding base e formulas (page 278), and again the simplicity of the base e formulas is why e is called the “natural” base. As before, these formulas reduce to the natural (base e ) formulas if a  e . 121–122. Use the formulas on the previous page to find the derivative of each function. 121. a. log 2 x b. log 10 ( x 2  1) c. log 3 ( x 4  2 x ) 122. a. log 3 x b. log 2 ( x 2  1) c. log 10 ( x 3  4 x ) 4.4 Two Applications to Economics: Relative Rates and Elasticity of Demand Introduction In this section we define relative rates of change and see how they are used in economics. ( Relative rates are not the same as the related rates discussed in Section 3.6.) We then define the very important economic concept of elasticity of demand. Relative Versus Absolute Rates The derivative of a function gives its rate of change.

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  Mathematical Applications for the Management, Life, and Social Sciences

  • Ronald Harshbarger, James J. Reynolds(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 11.1 Derivatives of Logarithmic Functions 693 Derivatives of Logarithmic Functions SEC TION 11.1 OBJECTIVES • To find derivatives of natural logarithmic functions • To find derivatives of logarithmic functions with bases other than e APPLICATION PREVIEW The table shows the expected life spans at birth for people born in certain years in the United States, with projections to 2050. These data can be modeled by the function l(x) 5 8.7744 1 14.907 ln x, where x is the number of years past 1900. The graph of this function is shown in Figure 11.1. If we wanted to use this model to find the rate of change of life span with respect to the number of years past 1900, we would need the derivative of this function and hence the derivative of the logarithmic function ln x. (See Example 4.) Year Life Span (years) Year Life Span (years) 1920 54.1 1990 75.4 1930 59.7 2000 76.8 1940 62.9 2010 78.7 1950 68.2 2020 80.2 1960 69.7 2030 81.7 1970 70.8 2040 83.0 1980 73.7 2050 84.4 Source: National Center for Health Statistics 0 160 50 90 l(x) = 8.7744 + 14.907 ln x Recall that we define the logarithmic function y 5 log a x as follows. Logarithmic Functions For a 7 0 and a notequal.alt1 1, the logarithmic function y 5 log a x (logarithmic form) has domain x 7 0, base a, and is defined by a y 5 x (exponential form) Logarithmic Function The a is called the base in both log a x 5 y and a y 5 x, and y is the logarithm in log a x 5 y and the exponent in a y 5 x.

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  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  350 Chapter 6 / Exponential, Logarithmic, and Inverse Trigonometric Functions LOGARITHMIC DIFFERENTIATION We now consider a technique called logarithmic differentiation that is useful for differen- tiating functions that are composed of products, quotients, and powers. Example 5 The derivative of y = x 2 3 √ 7x − 14 (1 + x 2 ) 4 (7) is messy to calculate directly. However, if we first take the natural logarithm of both sides and then use its properties, we can write ln y = 2 ln x + 1 3 ln(7x − 14) − 4ln(1 + x 2 ) Differentiating both sides with respect to x yields 1 y dy dx = 2 x + 7 / 3 7x − 14 − 8x 1 + x 2 Thus, on solving for dy /dx and using (7) we obtain dy dx = x 2 3 √ 7x − 14 (1 + x 2 ) 4  2 x + 1 3x − 6 − 8x 1 + x 2  REMARK Since ln y is only defined for y > 0, the computations in Example 5 are only valid for x > 2 (verify). However, using the fact that the expression for the derivative of ln y is the same as that for ln |y|, it can be shown that the formula obtained for dy / dx is valid for x < 2 as well as x > 2 (Exercise 75). In general, whenever a derivative dy / dx is obtained by logarithmic differentiation, the resulting derivative formula will be valid for all values of x for which y = 0. It may be valid at those points as well, but it is not guaranteed. INTEGRALS INVOLVING ln x Formula (2) states that the function ln x is an antiderivative of 1 / x on the interval (0, +∞), whereas Formula (6) states that the function ln |x| is an antiderivative of 1 / x on each of the intervals (−∞, 0) and (0, +∞). Thus we have the companion integration formula to (6),  1 u du = ln |u| + C (8) with the implicit understanding that this formula is applicable only across an interval that does not contain 0. Example 6 Applying Formula (8),  e 1 1 x dx = ln |x|] e 1 = ln |e| − ln |1| = 1 − 0 = 1  −1 −e 1 x dx = ln |x|] −1 −e = ln | −1| − ln | − e| = 0 − 1 = −1 Example 7 Evaluate  3x 2 x 3 + 5 dx.

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  Calculus, Volume 1

  • Tom M. Apostol(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  6.2 Motivation for the definition of the natural logarithm as an integral The logarithm is an example of a mathematical concept that can be defined in many different ways. When a mathematician tries to formulate a definition of a concept, such as the logarithm, he usually has in mind a number of properties he wants this concept to have. By examining these properties, he is often led to a simple formula or process that might serve as a definition from which all the desired properties spring forth as logical deductions. We shall illustrate how this procedure may be used to arrive at the definition of the logarithm which is given in the next section. One of the properties we want logarithms to have is that the logarithm of a product should be the sum of the logarithms of the individual factors. Let us consider this property by itself and see where it leads us. If we think of the logarithm as a function f , then we want this function to have the property expressed by the formula f (xy) = f (x) + f (y) (6.4) whenever x, y, and xy are in the domain of f . An equation like (6.4), which expresses a relationship between the values of a function at two or more points, is called a functional equation. Many mathematical problems can be reduced to solving a functional equation, a solution being any function which satisfies the equation. Ordinarily an equation of this sort has many different solutions, and it is usually very difficult to find them all. It is easier to seek only those solutions which have some additional property such as continuity or differentiability. For the most part, these are the only solutions we are interested in anyway. We shall adopt this point of view and determine all differentiable solutions of (6.4). But first let us try to deduce what information we can from (6.4) alone, without any further restrictions on f . One solution of (6.4) is the function that is zero everywhere on the real axis.

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  eBook - PDF

  Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  1, the integral represents the area of the shaded region in Figure 5. x 0 y 0.5 1 e y= ln x x FIGURE 5 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 6.4 Derivatives of Logarithmic Functions 445 Since ln b is a constant, we can differentiate as follows: d dx slog b xd - d dx ln x ln b - 1 ln b d dx sln xd - 1 x ln b 6 d dx slog b xd - 1 x ln b EXAMPLE 12 Differentiate f s xd - log 10 s2 1 sin xd. SOLUTION Using Formula 6 with b - 10 and the Chain Rule, we get f 9 s xd - d dx log 10 s2 1 sin xd - 1 s2 1 sin xd ln 10 d dx s2 1 sin xd - cos x s2 1 sin xd ln 10 ■ From Formula 6 we see one of the main reasons that natural logarithms (logarithms with base e) are used in calculus: the differentiation formula is simplest when b - e because ln e - 1. Exponential Functions with Base b In Section 6.2 we showed that the derivative of the general exponential function f s xd - b x , b . 0, is a constant multiple of itself: f 9 s xd - f 9 s0d b x where f 9 s0d - lim h l 0 b h 2 1 h We are now in a position to show that the value of the constant is f 9 s0d - ln b. 7 d dx sb x d - b x ln b PROOF We use the fact that e ln b - b: d dx sb x d - d dx se ln b d x - d dx e sln bd x - e sln bd x d dx sln bd x  - se ln b d x sln bd - b x ln b  ■ In Example 2.7.6 we considered a population of bacteria cells that doubles every hour and we saw that the population after t hours is n - n 0 2 t , where n 0 is the initial popula- tion. Formula 7 enables us to find the growth rate: dn dt - n 0 2 t ln 2 Copyright 2021 Cengage Learning.

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  Mathematics

  An Applied Approach

  • Michael Sullivan, Abe Mizrahi(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  That is, (6) d dx log a x  1 x ln a If f (x)  log a x, then f (x)  1 x ln a ▲ EXAMPLE 10 Finding the Derivative of log 2 x Find the derivative of: f (x)  log 2 x Using Formula (6), we have ◗ Formula (6) NOW WORK PROBLEM 47. f  (x)  d dx log 2 x  1 x ln 2 SOLUTION The Derivatives of the Exponential and Logarithmic Functions; the Chain Rule 755 To find the derivative of f (x)  a x , where a  0, a  1, is any real constant, we use the definition of a logarithm and the Change-of-Base Formula. If y  a x , we have x  log a y Definition of a logarithm. Apply the Change-of-Base Formula. Substitute y  a x . Now, we differentiate both sides with respect to x: ln a is a constant. Use Formula (5). Simplify. Solve for . We have derived the formula: d dx a x d dx a x  a x ln a 1  d dx a x a x ln a 1  1 ln a  d dx a x a x 1  1 ln a  d dx ln a x d dx x  d dx ln a x ln a x  ln a x ln a x  ln y ln a EXAMPLE 11 Finding the Derivative of 2 x Find the derivative of: f (x)  2 x Using Formula (7), we have ◗ Formula (7) NOW WORK PROBLEM 51. f  (x)  d dx 2 x  2 x ln 2 EXAMPLE 12 Maximizing Profit At a Notre Dame football weekend, the demand for game-day t-shirts is given by where p is the price of the shirt in dollars and x is the number of shirts demanded. p  30  5 ln  x 100  1  Derivative of f (x)  a x The derivative of f (x)  a x , a  0, a  1, is f (x)  a x ln a. That is, (7) d dx a x  a x ln a ▲ SOLUTION 756 Chapter 13 The Derivative of a Function (a) At what price can 1000 t-shirts be sold? (b) At what price can 5000 t-shirts be sold? (c) Find the marginal demand for 1000 t-shirts and interpret the answer. (d) Find the marginal demand for 5000 t-shirts and interpret the answer. (e) Find the revenue function R  R(x). (f) Find the marginal revenue from selling 1000 t-shirts and interpret the answer. (g) Find the marginal revenue from selling 5000 t-shirts and interpret the answer. (h) If each t-shirt costs $4, find the profit function P  P(x).

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  A Course of Mathematical Analysis

  International Series of Monographs on Pure and Applied Mathematics

  • A. F. Bermant, I. N. Sneddon, S. Ulam, M. Stark(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  DERIVATIVES AND DIFFERENTIALS 157 We have: (sinh#) = = = cosh#, Δ Δ (cosh#)' = = = sinh#, Δ Δ sinh x ' (sinh x)' cosh x — (cosh x)' sinh x V cosh a: / cosh 2 x cosh 2 # — sinh 2 a; 1 cosh 2 # cosh 2 a; ' A similarity can again be noticed here between the hyperbolic and trigonometric functions. 49. Logarithmic differentiation. Differentiation of inverse and im-plicit functions. I. By using the rules for differentiation and the table of derivatives of the basic elementary functions, we can now find automatically the derivatives of any elementary functions, except for one type, the simplest representative of which is the function y = x*. Such functions are described as power-exponential and include, in general, any function written as a power whose base and index both depend on the independent variable. In order to find by the general rules the derivative of the power-exponential function y = x?, we take logarithms of both sides of this equation: _ my = # l n # , x > 0. Since this is an identity, the derivative of the left-hand side must be equal to the derivative of the right. We obtain by differentiating with respect to x (and not forgetting that the left-hand side is a function of a function): 1 / i — y = In x -f- 1. y Hence y' = y(mx + 1) = a* (In a; + 1). The operation consisting in first taking the logarithm of the func-tion / (x) (to base e) then differentiating is called logarithmic differen-tiation and its result ,, is called the logarithmic derivative of f(x). 158 COURSE OF MATHEMATICAL ANALYSIS Logarithmic differentiation can be used for finding other derivatives besides those of power-exponential functions; it can make for shorter working in these other cases. For instance, we can use logarithmic differentiation to find the derivative of y= ]/x* + 4 sin 2 x 2 X more rapidly. We have: In y — In (x 2 + 4) + In sin 2 x + x In 2, 2/ ' = y (^TT + 2cota: + ln2 )· II.

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