Mathematics
Derivative of Exponential Function
The derivative of an exponential function, such as f(x) = e^x, is itself. In other words, the rate of change of an exponential function at any point is equal to the value of the function at that point. This property makes exponential functions unique and important in calculus and mathematical modeling.
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5 Key excerpts on "Derivative of Exponential Function"
- Ron Larson(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Use calculus to analyze the graphs of real-life functions that involve the natural exponential function. Explore normal probability density functions. Derivatives of Exponential Functions In Section 4.2, it was stated that the most convenient base for exponential functions is the irrational number e . The convenience of this base stems primarily from the fact that the function f ( x ) = e x is its own derivative . You will see that this is not true of other exponential functions of the form y = a x where a ≠ e . To verify that f ( x ) = e x is its own derivative, notice that the limit lim Δ x uni2192 0 ( 1 + Δ x ) 1 H20862Δ x = e implies that for small values of Δ x , e ≈ ( 1 + Δ x ) 1 H20862Δ x or e Δ x ≈ 1 + Δ x . This approximation is used in the following derivation. f uni2032 ( x ) = lim Δ x uni2192 0 f ( x + Δ x ) -f ( x ) Δ x Definition of derivative = lim Δ x uni2192 0 e x +Δ x -e x Δ x Use f ( x ) = e x . = lim Δ x uni2192 0 e x ( e Δ x -1 ) Δ x Factor numerator. = lim Δ x uni2192 0 e x [( 1 + Δ x ) -1 ] Δ x Substitute 1 + Δ x for e Δ x . = lim Δ x uni2192 0 e x ( Δ x ) Δ x Divide out common factor. = lim Δ x uni2192 0 e x Simplify. = e x Evaluate limit. When u is a differentiable function of x , you can apply the Chain Rule to obtain the derivative of e u with respect to x . Both formulas are summarized below. Derivative of the Natural Exponential Function Let u be a differentiable function of x . 1. d dx [ e x ] = e x 2. d dx [ e u ] = e u du dx In Exercise 48 on page 274, you will use the derivative of an exponential function to find the rate of change of the average typing speed after 5, 10, and 30 weeks of lessons. Blaj Gabriel/Shutterstock.com Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300
eBook - PDFThe Calculus Lifesaver
All the Tools You Need to Excel at Calculus
- Adrian Banner(Author)
- 2009(Publication Date)
- Princeton University Press(Publisher)
Here are a couple of simple examples of using the formula. Working on the domain (0 , ∞ ), what is the derivative of x √ 2 with respect to x ? How about x π ? Just use the formula to show that d dx ( x √ 2 ) = √ 2 x √ 2 -1 and d dx ( x π ) = πx π -1 for x > 0. It’s not really any different from what we’ve done before—just that we can handle non-integer exponents now. 9.6 Exponential Growth and Decay We’ve seen that bank accounts with continuous compounding grow exponen-tially. We don’t need to look to such human-made devices to find exponen-tial growth, though: it occurs in nature too. For example, under certain circumstances, populations of animals, like rabbits (and humans!), grow ex-ponentially. There’s also exponential decay, where a quantity gets smaller and smaller in an exponential fashion (we’ll see what this means very soon). This occurs in radioactive decay, allowing scientists to find out how old some ancient artifacts, fossils, or rocks are. Here’s the basic idea. Suppose y = e kx . Then, as we saw at the beginning of Section 9.3.1 above, dy/dx = ke kx . The right-hand side of this equation can be written as ky , since y = e kx . That is, dy dx = ky. This is an example of a differential equation . After all, it’s an equation involv-ing derivatives. We’ll look at many more differential equations in Chapter 30, but let’s just focus on this one for the moment. What other functions satisfy the above equation? We know that y = e kx does, but there must be others. For example, if y = 2 e kx , then dy/dx = 2 ke kx , which is once again equal to ky . More generally, if y = Ae kx , then dy/dx = Ake kx , which is once again equal to ky . It turns out that this is the only way you can have dy/dx = ky : if dy dx = ky, then y = Ae kx for some constant A. We’ll see why in Section 30.2 of Chapter 30. In the meantime, let’s take a closer look at the differential equation dy/dx = ky .
CHAPTER 6 The Differential Equation of Intrinsic Growth: the Exponential and Logarithmic Functions INTRODUCTION The exponential functions are prominent in mathematical science, partly because they describe a certain natural pattern of growth or decay. Analysis of that pattern will lead in this chapter to a definition of the functions. The first article is devoted to giving the concept of organic growth the form of a differential equation and investigating the nature of the solutions of that equation, by which growth must be represented. In the second article the equation is solved and the desired functions are defined together with the logarithms. A number of applications of these ubiquitous functions then follow. I will usually allow “growth,” “increase,” etc., to stand for “growth (or decay),” “increase (or decrease),” etc. The other case should be mentally supplied, as can be done by a straightforward analogy. 6.1 INTRINSIC GROWTH 6.1.1 Extrinsic and intrinsic uniform growth. Let a homogeneous substance increase continuously in quantity—mass, volume, or some other mea-sure. The growth is uniform , as we understood the term in Art. 1.2, if it is as described in § 3.6.5; in particular, if either of the two following equivalent conditions ((iii) and (iv) from that section) is satisfied. (1) In any equal periods of time the substance is increased by addition of the same quantity. (2) The rate of increase (increase per time) is constant. 304 § 6.1.1 INTRINSIC GROWTH 305 Denoting the rate of increase by a and the quantity present at time t by u ( t ) we can express the second of these by the differential equation 1 du dt = a . Integration shows at once that its solutions are the linear functions of the form u ( t ) = u 0 + at , where u 0 = u (0), and a process so described is often called linear growth. As an example of this kind of growth I have used the increase of water in a vessel being filled by a steady stream.
eBook - PDF- Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
CHAPTER 5 Growth and Decay: An Introduction to Exponential Functions Overview Exponential and linear functions are used to describe quantities that change over time. Expo- nential functions represent quantities that are multiplied by a constant factor during each time period. Linear functions represent quantities to which a fixed amount is added (or subtracted) during each time period. Exponential functions can model such diverse phenomena as bacteria growth, radioactive decay, compound interest rates, inflation, musical pitch, and family trees. After reading this chapter, you should be able to • recognize the properties of exponential functions and their graphs • understand the differences between exponential and linear growth • model growth and decay phenomena with exponential functions • represent exponential functions using percentages, factors, or rates • use semi-log plots to determine if data can be modeled by an exponential function • use base e to understand and model continuous or instantaneous compounding 255 5.1 Exponential Growth The Growth of E. coli Bacteria Measuring and predicting growth is of concern to population biologists, ecologists, demogra- phers, economists, and politicians alike. The growth of bacteria provides a simple model that scientists can generalize to describe the growth of other phenomena such as cells, countries, or capital. Bacteria are very tiny, single-celled organisms that are by far the most numerous organ- isms on Earth. One of the most frequently studied bacteria is E. coli, a rod-shaped bacterium approximately 10 −6 meter (or 1 micrometer) long that inhabits the intestinal tracts of humans and other mammals. 1 The cells of E. coli reproduce by a process called fission: The cell splits in half, forming two “daughter cells.” See course software “E2: Exponential Growth & Decay” for a dramatic visualization of growth and decay. Applet 1 Most types of E. coli are beneficial to humans, aiding in digestion.
eBook - PDFBiocalculus
Calculus, Probability, and Statistics for the Life Sciences
- James Stewart, Troy Day, James Stewart(Authors)
- 2015(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. 216 CHAPTER 3 | Derivatives actually happens fairly accurately. Another example occurs in nuclear physics where the mass of a radioactive substance decays at a rate proportional to the mass. In chemistry, the rate of a unimolecular first-order reaction is proportional to the concentration of the substance. In general, if y s t d is the value of a quantity y at time t and if the rate of change of y with respect to t is proportional to its size y s t d at any time, then (1) dy dt -ky where k is a constant. Equation 1 is sometimes called the law of natural growth (if k . 0 ) or the law of natural decay (if k , 0 ). It is called a differential equation because it involves an unknown function y and its derivative dy y dt . It’s not hard to think of a solution of Equation 1. This equation asks us to find a func-tion whose derivative is a constant multiple of itself. We have met such functions in this chapter. Any exponential function of the form y s t d -Ce kt , where C is a constant, satisfies y 9 s t d -C s ke kt d -k s Ce kt d -ky s t d We will see in Section 7.4 that any function that satisfies dy y dt -ky must be of the form y -Ce kt . To see the significance of the constant C , we observe that y s 0 d -Ce k ? 0 -C Therefore C is the initial value of the function.
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