Derivative

“ 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. ”

“ 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. ”

“ 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 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. ”

“ By continuing the process as illustrated in I, II, and III, it is seen that if the first derivative of which does not vanish for x = a is of even order (= n) , then (47) is a maximum if a negative number; (48) is a minimum if a positive number. [3] If the first derivative of which does not vanish for x = a is of odd order, then will be neither a maximum nor a minimum. ”

“ 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. ”

“ 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. ”

“ 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 ”

“ 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. ”

“ 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. ”

“ The more a person would be able to use the abstract instead of the original work, the more likely the abstract would be deemed a derivative work. The longer and more comprehensive the abstract is, the greater the chance it will be considered a derivative work. But even a short abstract still distills the essence of the original work—one which can substitute quite well for the original work—may also be considered a derivative work. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ On Incompletion of the Radical.—Let there be in a given language a series of roots ending in -t, as sæmat. Let a euphonic influence eject the -t, as often as the word occurs in the nominative case. Let the nominative case be erroneously considered to represent the root, or radical, of the word. Let a derivative word be formed accordingly, i.e., on the notion that the nominative form and the radical form coincide. Such a derivative will exhibit only a part of the root; in other words, the radical will be incomplete. ”

“ 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 ”

“ This example is a striking instance of the fact that the termination óg is losing (if it has not already lost) its diminutive force.
All derived nouns in óg are feminine.
Derived Nouns.
461. Words are of three classes—Simple, Derivative, and Compound. All simple words are, as a general rule, monosyllables; they are the roots from which derivative and compound words spring. Derivative words are made up of two or more parts. These parts undergo slight changes when they are united to form words, and thus the component parts are somewhat disguised. ”

“ The languages of the third period of evolution, Latin for example, present a considerable number of these secondary and tertiary derivatives. The Latin word pater, father, is a primary derivative, of which the full or radical element is pa and the limiting element is ter. Paternus, whence our paternal, is a secondary, and paternitas, corresponding with our paternity, is a tertiary derivative. But our languages have not the extraordinary facility in derivation possessed by some simply agglutinative idioms. ”

“ All words are either primitive or derivative. A primitive word is that which cannot be reduced to any simpler word in the language; as, brass, York, knave. A derivative word, under the head of which compound words are also included, is that which may be reduced to another and a more simple word in the English language ”

“ Not derivatives of every kind are thus made; but, in the main, those classes which have most of the verbal force, or which are most akin in value with infinitives and participles.
c. The occurrence of such derivatives with prefixes, and their accent, will be noted under each suffix below. They are chiefly (in nearly the order of their comparative frequency) , besides root-stems, those in a, in ana, in ti, in tar and tra, and in in, ya, van and man, i and u, as, and a few others. ”

“ The same series of derivative verbs occurs in G. and R. as in H. Thus, we have a potential passive (a simple passive in G.) formed by adding ā to the base, as in G. lakhavũ, to write, lakhāvũ, to be written; and a causal by adding āv or āḍ, as in lakhāvavũ, to cause to write; besavũ, to sit, besāḍavũ, to seat. ”

“ No one doubts that thought, at least reflective as distinct from what is sometimes called constitutive thought, is derivative and secondary. It comes after something and out of something, and for the sake of something. ”

“ 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. ”

“ 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 ”

“ Such is the exuberance of signification which many words have obtained, that it was scarcely possible to collect all their senses; sometimes the meaning of derivatives must be sought in the mother term, and sometimes deficient explanations of the primitive may be supplied in the train of derivation. In any case of doubt or difficulty, it will be always proper to examine all the words of the same race ”

“ 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? ”

“ Since the formulas deduced in the Differential Calculus generally involve derivatives of y with respect to x, such formulas as (A) and (B) are especially useful when the parametric equations of a curve are given. ”

“ Over one thousand.
52. What is a Prefix?
That part of a derivative word placed before the root.
53. What is a Postfix?
That part of a derivative word placed after the root.
54. What is a Suffix?
Same as a postfix.
55. What are Affixes?
Prefixes and postfixes together are called affixes.
56. How many kinds of Derivatives are there?
Two.
57. What are they?
Regular and irregular.
58. What is a Regular derivative?
One that is formed by the addition of affixes without changing the letters in the primitive part (except final e silent) . ”