Derivative work

“ It is not indeed unusual that different ultimate properties should, in some particular instances, produce the same derivative property; but on the whole it is certain that things which differ in their ultimate properties, will differ at least as much in the aggregate of their derivative properties, and that the differences which are unknown will on the average of cases bear some proportion to those which are known. There will, therefore, be a competition between the known points of agreement and the known points of difference in A and B ”

“ The preceding considerations have led us to recognize a distinction between two kinds of laws, or observed uniformities in nature: ultimate laws, and what may be termed derivative laws. Derivative laws are such as are deducible from, and may, in any of the modes which we have pointed out, be resolved into, other and more general ones. ”

“ Let us designate the derivative function by the letter δ placed before the primitive function. The derivation indicated by this letter will depend upon the multiplier of V, which we will call T and which we will suppose developed like V by the ratio to the powers of the variable t. If we multiply anew by T the product of V by T, which is equivalent to multiplying V by T2, we shall form a third discriminant function, in which the coefficient of the xth power of t will be a derivative similar to the corresponding coefficient of the preceding product ”

“ From the Theory of Limits it is clear that if the derivative of a function exists for a certain value of the independent variable, the function itself must be continuous for that value of the variable. ”

“ The nth derivative. For certain functions a general expression involving n may be found for the nth derivative. The usual plan is to find a number of the first successive derivatives, as many as may be necessary to discover their law of formation, and then by induction write down the nth derivative. ”

“ The first derivative gives us the slope of the curve at any point; the second derivative determines the intervals within which the curve is concave upward or concave downward, and the points of inflection separate these intervals; the maximum points are the high points and the minimum points are the low points on the curve. ”

“ There are lots of ways to handle derivative works. Oldwikisource:Wikisource:Subdomain coordination gives a look at some of the different approaches. Hebrew Wikisource, for example, has a separate namespace for derivative works; the only example of this in any Wikisource as far as I am aware. On the other hand, Italian Wikisource bans both translation and annotation (and, presumably, comparison or any other form of "value-added" text) . We could go either way or somewhere in between. - AdamBMorgan (talk) 19:55, 13 February 2013 (UTC)
Copyvio I am afraid we have to delete St. ”

“ To classify derivations in this manner is to classify them according to their form. For the detail of the derivative forms, see Deutsche Grammatik, ii. 89-405.
IV. According to the historical origin of the derivational elements.—For this see the Chapter upon Hybridism.
V. According to the number of the derivational elements.—In fisher, as compared with fish, there is but one derivational affix. In fishery, as compared with fish, the number of derivational elements is two.
§ 427. The list (taken from Walker) of words alluded to in p. 293, is as follows:—
§ 428. ”

“ How distinguish between an affix and a part of a compound word?
If all the parts retain their literal signification they form a compound; if not, the part which loses its signification becomes an affix in a derivative.
72. Is the word outside compound or derivative?
It is compound.
73. Is the word outrun compound or derivative?
It is derivative.
74. What is Derivation?
That branch of etymology which treats of the sources of the words of a language. ”

“ All admit that the concept of a derivative is one of the most useful elementary tools of thought, and in a number of countries this concept has been introduced into secondary mathematics and used with success. ”

“ Infinitesimals. In the Differential Calculus we are usually concerned with the derivative, that is, with the ratio of the differentials dy and dx. In some applications it is also useful to consider dx as an infinitesimal (see §15) , that is, as a variable whose values remain numerically small, and which, at some stage of the investigation, approaches the limit zero. ”

“ A derivative law which results wholly from the operation of some one cause, will be as universally true as the laws of the cause itself; that is, it will always be true except where some one of those effects of the cause, on which the derivative law depends, is defeated by a counteracting cause. But when the derivative law results not from different effects of one cause, but from effects of several causes, we can not be certain that it will be true under any variation in the mode of co-existence of those causes, or of the primitive natural agents on which the causes ultimately depend. ”

“ Since the differential of a function is its derivative multiplied by the differential of the independent variable, it follows at once that the formulas for finding differentials are the same as those for finding derivatives given in § 33, if we multiply each one by dx. ”

“ The infinitely small increment of a function is in general of the same order as the corresponding increment of the variable, that is to say, their ratio has a finite limit.
Definitions of the derivative and differential of a function of a single variable. Tangents and normals to plane curves, whose equation in linear or polar coordinates is given.
A function is increasing or decreasing, according as its derivative is positive or negative. If the derivative is zero for all values of the variable, the function is constant. ”

“ His chief work, the Calcul des Dérivations, 1800, gives the method known by his name, by which the successive coefficients of a development are derived from one another when the expression is complicated. De Morgan has pointed out that the true nature of derivation is differentiation accompanied by integration. In this book for the first time are the symbols of operation separated from those of quantity. The notation for is due to him. ”

“ If we differentiate with respect to , regarding as constant, and then this result with respect to , regarding as constant, we obtain
which we denote by
Similarly, if we differentiate twice with respect to and then once with respect to , the result is denoted by the symbol
129.Order of differentiation immaterial. Consider the function . Changing into and keeping constant, we get from the Theorem of Mean Value, (46) , § 106,
(A) and since varies while remains constant, we get the partial derivative with respect to . ”

“ If, then, it is only to cases which in point of time are adjacent (or nearly adjacent) to those which we have actually observed, that any derivative law, not of causation, can be extended with an assurance equivalent to certainty, much more is this true of a merely empirical law. Happily, for the purposes of life it is to such cases alone that we can almost ever have occasion to extend them. ”

“ If, after much observation of B, we find that it agrees with A in nine out of ten of its known properties, we may conclude with a probability of nine to one, that it will possess any given derivative property of A. If we discover, for example, an unknown animal or plant, resembling closely some known one in the greater number of the properties we observe in it, but differing in some few, we may reasonably expect to find in the unobserved remainder of its properties, a general agreement with those of the former ”

“ Derivation, as a topic to be treated by the grammarian, is a species of Etymology, which explains the various methods by which those derivative words which are not formed by mere grammatical inflections, are deduced from their primitives. Most of those words which are regarded as primitives in English, may be traced to ulterior sources, and many of them are found to be compounds or derivatives in the other languages from which they have come to us. ”

“ The treatment of the Latin derivatives in Part II. presents a new and important feature, to wit: the systematic analysis of the structure and organism of derivative words, together with the statement of their primary meaning in such form that the pupil inevitably perceives its relation with the root, and in fact makes its primary meaning by the very process of analyzing the word into its primitive and its modifying prefix or suffix. ”

“ All the notes of the scale have now been displaced.
In like manner, the displacement downward of one and only one note (B) of the diatonic order results in the same sequence of tones and hemitones. The B of this first derivative is the E of the original order, the displacement of which gives a second derivative; as before, the displacement downward of E is impossible without that of B. The third to the seventh derivatives are formed by the additional displacements downward of A, D, G, C, F, respectively, that of each demanding that of the previous note of the series. ”

“ Whatever, namely, may have been originally and strictly the mode of production of the derivatives with prefixes, it is throughout the recorded life of the language as if the root and its prefix or prefixes constituted a unity, from which a derivative is formed in the same manner as from the simple root, with that modification of the radical meaning which appears also in the proper verbal forms as compounded with the same prefixes. ”

“ This function assumes the indeterminate form for . Let then when . By § 113, when . By § 112, when x = 0.
EXAMPLES
Evaluate the following expressions by differentiation:
1. Ans. 7. Ans. 2. 1. 8. 3. 1. 9. 4. 10. 5. e. 11. 6. 12. 1. 13. ↑ Also called the Law of the Mean. ↑ Also called the Extended Law of the Mean. ↑ As in § 82, a critical value x = a is found by placing the first derivative equal to zero and solving the resulting equation for real roots. ↑ x = 0 is the only root of the equation . ↑ The calculation of this limiting value is called evaluating the indeterminate form. ”

“ Derivative classifiers shall, whenever practicable, use a classified addendum whenever classified information constitutes a small portion of an otherwise unclassified document or prepare a product to allow for dissemination at the lowest level of classification possible or in unclassified form. (d) Persons who apply derivative classification markings shall receive training in the proper application of the derivative classification principles of the order, with an emphasis on avoiding over-classification, at least once every 2 years. ”

“ As experience increases the regularity of the action of the derivative force is more and more observable, and then arises the notion of a law or rule regulating the action of every such force. And a perpetually increasing number of phenomena are brought under this head, and are shown to be, not the immediate results of self-originating action, but the more or less remote results of derivative action governed by laws. ”

“ The derivatives I have referred to their primitives, with an accuracy sometimes needless; for who does not see that remoteness comes from remote, lovely from love, concavity from concave, and demonstrative from demonstrate? But this grammatical exuberance the scheme of my work did not allow me to repress. It is of great importance, in examining the general fabrick of a language, to trace one word from another, by noting the usual modes of derivation and inflection; and uniformity must be preserved in systematical works, though sometimes at the expense of particular propriety. ”

“ Or we can say, and it is just the same thing in other words:
(speed) equals the derivative of (distance) with respect to (time)
Now we can tell what a differential equation is. It is simply an equation in which a derivative occurs, such as the last example. Perhaps the commonest kind of equation in physical problems is the differential equation.
SOLVING PHYSICAL PROBLEMS
Now we were able to change the equation about floor area into other forms, if we wanted to find length or width instead of floor area. When we did this, we ran into the inverse or opposite of multiplication: division. ”

“ The new ideal entities are called Derived Functions, or Differential Coefficients, and for a long time it was assumed as almost an axiom that every function continuous within given intervals must have, within those intervals, a derived function, or differential coefficient. ”

“ In order to determine sufficient conditions that shall be a maximum or a minimum, it is necessary to proceed to higher derivatives. To derive sufficient conditions for all cases is beyond the scope of this book. ”

“ A Derivative Law will be true wherever the forces concerned exist in the combinations upon which the law depends, if there are no counteracting conditions. That water can be pumped to about 33 feet at the sea-level, is a derivative law on this planet: is it true in Mars? ”

“ We are disposed to rank the derivative hypothesis in its fullness with the nebular hypothesis, and to regard both as allowable, as not unlikely to prove tenable in spite of some strong objections, but as not therefore demonstrably true. Those, if any there be, who regard the derivative hypothesis as satisfactorily proved, must have loose notions as to what proof is. Those who imagine it can be easily refuted and cast aside, must, we think, have imperfect or very prejudiced conceptions of the facts concerned and of the questions at issue. ”