Mathematics
Higher-Order Derivatives
Higher-order derivatives refer to the derivatives of a function beyond the first derivative. They represent the rate of change of the rate of change, and so on, of the original function. For example, the second derivative represents the curvature of the function, while the third derivative represents the rate of change of curvature, and so forth.
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6 Key excerpts on "Higher-Order Derivatives"
eBook - PDF- David Pearson(Author)
- 1995(Publication Date)
- Butterworth-Heinemann(Publisher)
There are other examples of functions for which, at various points, derivatives up to some order k exist but the (k + I )th derivative does not. These cases show that, even for differentiable functions, there is no guarantee that higher order derivatives can be defined at all points in their domain. One of the important applications of higher order derivatives is to power series. Letfbe a function which has a power series expansion, convergent within its radius of convergence. Then (-R < x < R) We know that such a power series can be differentiated term by term, provided we keep within the prescribed interval Ixl < R. Moreover, the differentiated series which is convergent with the same radius of convergence R, can again be differentiated to obtain a series for the second derivative off This process can be continued to give power series expansions for all higher order derivatives of f, again convergent within the same interval. We can list these series, together with Finding Out About Functions 69 the original series for f, as follows: f(x) == ao + atX + a2x2 + a3 x3 + . f'(x) == at + 2a2x + 3a3x2 + . f (x) == 2a2 + 3. 2a3x + . f'(x) == 3. 2a 3 + . ... , and so on The situation is particularly simple at x == 0, where we havef(O) == ao,f'(O) == aI, f(O) == 2a2, f'''(O) == 3. 2a3, ..., and in general f(n) (0) == n!a n. Hence, an == f(n)(O)/n!, and this result enables us to determine all the coefficients in the power series for f(x) , given the values off, f', and all higher order derivatives, at the single point x == O. Summarizing this information, we see that the power series forf(x) may be written as 2 3 ... .n ( ) _ ( ) '() x () x '() A (n) ( ) f x -f 0 + xf 0 + 2! I 0 + 3T I 0 + ... + n! f 0 + ... (6.1) and is convergent within the radius of convergence R. Of course, not all functions may be expanded in a power series in this way.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
It is still a function, but its domain is strictly smaller than the domain of ƒ . Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by D , then D ( ƒ ) is the function f′ ( x ). Since D ( ƒ ) is a function, it can be evaluated at a point a . By the definition of the derivative function, D ( ƒ )( a ) = f′ ( a ). For comparison, consider the doubling function ƒ ( x ) =2 x ; ƒ is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs: The operator D , however, is not defined on individual numbers. It is only defined on functions: Because the output of D is a function, the output of D can be evaluated at a point. For instance, when D is applied to the squaring function, D outputs the doubling function, which we named ƒ ( x ). This output function can then be evaluated to get ƒ (1) = 2, ƒ (2) = 4, and so on. ________________________ WORLD TECHNOLOGIES ________________________ Higher derivatives Let ƒ be a differentiable function, and let f′(x) be its derivative. The derivative of f′(x) (if it has one) is written f′′(x) and is called the second derivative of ƒ . Similarly, the derivative of a second derivative, if it exists, is written f′′′(x) and is called the third derivative of ƒ . These repeated derivatives are called Higher-Order Derivatives . A function ƒ need not have a derivative, for example, if it is not continuous. Similarly, even if ƒ does have a derivative, it may not have a second derivative. For example, let Calculation shows that ƒ is a differentiable function whose derivative is f′(x) is twice the absolute value function, and it does not have a derivative at zero. Similar examples show that a function can have k derivatives for any non-negative integer k but no (k + 1) -order derivative.
eBook - PDFPhysical Oceanography
A Mathematical Introduction with MATLAB
- Reza Malek-Madani(Author)
- 2012(Publication Date)
- Chapman and Hall/CRC(Publisher)
Higher order derivatives of f are defined analogously, by applying the formulas in (3.1)–(3.4) to lower order derivatives. For example, f primeprime ( a ) is determined as f primeprime ( a ) = lim h → 0 f prime ( a + h ) -f prime ( a ) h . (3.5) We may apply any of the formulas in (3.1)–(3.4) to the right side of (3.5) to arrive at formulas for f primeprime ( a ) that involve evaluation of the function f at a and its neighboring values and not any of its derivatives. For instance f primeprime ( a ) = lim h → 0 f ( a + h ) -2 f ( a ) + f ( a -h ) h 2 (3.6) and f primeprime ( a ) = lim h → 0 f ( a + 2 h ) -2 f ( a + h ) + f ( a ) h 2 (3.7) result from applying (3.1) and (3.2) (see the Problem 8 at the end of this section). Partial derivatives of a function f , when f depends on several in-dependent variables, are defined precisely as laid out in (3.1) and its equivalent forms because the partial derivative of f with respect to one of its independent variables, say x , is the rate of change of f when x is allowed to vary while all other independent variables are kept constant. For simplicity, let us assume f is a function of three variables, denoted by x , y and z , in a domain D , a subset of R 3 . Let P = ( a,b,c ) be a point in the domain at which we are interested in determining f ’s rate of change in the x direction. This quantity, which we denote by ∂f ∂x or by f x , is obtained as follows: ∂f ∂x | P = lim h → 0 f ( a + h,b,c ) -f ( a,b,c ) h , (3.8) if the limit exists. We obtain ∂f ∂y and ∂f ∂z in a similar fashion: ∂f ∂y | P = lim h → 0 f ( a,b + h,c ) -f ( a,b,c ) h , ∂f ∂z | P = lim h → 0 f ( a,b,c + h ) -f ( a,b,c ) h . (3.9) Higher order derivatives of f are obtained by repeated application of the definition of differentiation. So, f xx | P is obtained by first computing f x
eBook - PDF- William Cox(Author)
- 1998(Publication Date)
- Butterworth-Heinemann(Publisher)
Apart from this it is simply a matter of notation, which again is fairly self-evident. For example, a::{x == :y (:):~) ==fxyy a~rl== a~(:A:y(;~))==hyxx and so on. And again, provided all appropriate derivatives are continuous on an open set, then everywhere on that open set the order of the differentiation is irrelevant -for example, L: = fxxy = t.; In practice, there is no great call for derivatives higher than second order -as you might suspect from the number of times you have had to deal with ordinary deriva-tives of greater than second order. A notable exception occurs in the theory of elas-ticity, where the biharmonic equation contains fourth-order derivatives, but that is perhaps stretching things a bit. EXERCISES ON 4.2 1. Find all first-and second-order partial derivatives, checking the equality of the mixed derivatives, for the following functions: (i) X 3 y2 + 4 xy4 (ii) In(x 2 + y2) (iii) e XY cos(x + y) ',> 2. Show thatf(x, y) = In(x 2 + y2) satisfies the partial differential equation alj alj -+-=0 ax 2 a y 2 This is called the Laplace equation in two-dimensional rectangular coordinates. 44 Vector Calculus It is very important in fluid mechanics, electromagnetism and many other areas of science and engineering, as well as being a key equation in pure mathematics. 4.3 Differentiation of functions of more than two variables I wonder if, like me, you find diagrams such as Fig. 1.2 less than enlightening? Certainly, such geometrical representations do not appeal to everyone. But they are clearly important, and they certainly give meaning to the idea of a derivative of a single variable, for example. But we should not forget that they are only diagram-matic illustrations of concepts which are not necessarily intrinsically geometric. For example, the slope of a curve is basically a geometric representation of a means of measuring the rate at which something is increasing.
eBook - PDFA 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)
194 COURSE OF MATHEMATICAL ANALYSIS Example. Let us find the hundredth derivative of the function y = x 2 sin a;. We have: 2/(100) = (a-2 s i n ^ ( ΐ ο ο ) = 100 · 99 = (sins)* 100 )* 2 + 100(sina;)< 99 > (x 2 )' + ^ (sina;) 98 (a 2 ); there is no need to work out the following terms since they are all zero: each contains as a factor a derivative of x 2 of higher than the second order. Hence (see Sec. 59), o;(ioo) = x 2 s i n L + loo -^ + 200 x sin (x + 99 + 9900 sin (x + 98 -^ ) = a; 2 sin a; - 200 a? cos a - 9900 sin x. 61· Differentials of higher orders. The differential dy of the func-tion y = f (x) is a function of two variables (Sec. 52): the independent variable x and its differential dx. The differential d# of the independent variable x is regarded as a magnitude independent of x: we can quote values of dx without taking account of the actual value of x. Let us take the differential d(df(x)) of df(x) as a function of x, i.e. the principal part of the increment df(x + dx) — df(x), proportional to dx. If this differential exists, it is called the second order or second differential of function f(x) and is written as d 2 y: d 2 y = d(dy). Similarly, the third differential d 3 y of f(x) is the differential of the second differential as a function of x. Let us give a general definition. Definition. The n-th differential d n y is the differential of the (n — l)-th differential as a function of xi d n y = d(d nl y). Let us now find an expression for the higher order differentials of function /, on the assumption that its argument is the independent variable x. We have for the second differential: d(dy) = (dy)' dx = [f (x) dx]' dx, DERIVATIVES AND DIFFERENTIALS 195 and since, by what has been said, dx must be regarded as a constant when differentiating with respect to x, we have d 2 y = f(x) dx-dx = f(x) dx 2 .
eBook - PDFA Course of Higher Mathematics
Adiwes International Series in Mathematics, Volume 1
- V. I. Smirnov, A. J. Lohwater(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
55] DIFFERENTIALS OF HIGHER ORDERS 125 We take the example of a point M oscillating harmonically on a line, so that its distance s from a fixed point 0 of the line is defined by the formula: 5 = asin + , where the amplitude a, the period of oscillation r, and the phase are constants. Differentiations give us the velocity v and the force /: 2πα COS ί_^£^ ω 1 f — mw== 4TI 2 m . ( 2π , , 4 2 m -a sin j 1 + OJ i.e. the force is proportionalen magnitude to the length of the interval OM and acts in the opposite direction. In other words, the force is always directed from the point M to the point 0, being proportional to their distance apart. 55. Differentials of higher orders. We now introduce the concept of higher order differentials of a function y = f(x) . Its differential ày = f'{x)&x is clearly a function of x, though it must be remembered that the differential d# of the independent variable is reckoned as independent of x [50], so that it must be taken outside the differentiation sign as a constant factor on further differentiation. The differential of ay can be obtained by treating it as a function of x; this is called the second order differential of the original function f(x) and is denoted by d 2 y or d 2 f(x): d?y = d(dy) -J'{x)dz]' dx =f(x)dx 2 . On obtaining the differential of this further function of x, we arrive at the third order differential: d 3 y = d(d 2 y) = [f(x)dx 2 ]' dx = f // (x)dx 2 , and in general, we arrive by successive differentiation at the concep: of the nth order differential of function/(#), which is expressed ast d n f(x) or d n y=f n) (x)dx n . (2) This formula allows of the expression of the nth derivative as a fraction: 126 DIFFERENTIATION: THEORY AND APPLICATIONS [56 We now consider a function of a function, y = f(u), where u is a function of some independent variable. We know [50] that the first differential of this function has the same form as when u is the inde-pendent variable: dy =f(u)du.
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