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dy = rf^jot-i- = -=^. (36) (»j) y = cosec-1 ^,

rf« = «f.cosec-1- = ;. (37)

In respect of the above differentials it is to be observed, that

00 00

the sum of d.sin-1- and </.cos-1- is zero: which arises from a a


the fact that the sum of two such arcs is ^, which is constant, and whose differential therefore is equal to zero. The same

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thing, and for the same reason, is true of tf.tan-1 - + d.cot-1 -,

a a

x x and of tf.sec-1 - 4- d.cosec-1 -.

a a

43.] Hence, if the radius is unity, we have

... dx , , —dx = -, a.co8-1# =

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let us take the Napierian logarithms of both members of the equation; and we have

logy = sin-1.r log«r.

dy , dx war^x.dx

= log x H ,

y 6 {l-x*}* *

.-. dy = x'm *\ 6 \ dx.

* I x(\-x*)b S

The number of examples which have been differentiated, including the types of all known algebraical, exponential, logarithmic, and circular forms, and all of which indicate the derived function to be a new function of x, is sufficient to justify the presumption of Art. 18, that the differential of f(x) is a product of two factors, of which one is dx and the other is a finite function of x. It is in fact almost an inductive argument per simplicem enumerationem.

Section 2.—The differentiation of functions of many variables. '16.] On functions of many variables.

When many variables are involved in an equation, it is theoretically possible so to arrange them that one shall be a function of all the others, whereby it becomes an explicit function of many variables. And these latter variables may be independent of each other, or have such relations amongst them that a variation of one may involve a variation of one or more of the others, whereby the number of independent variables may be diminished; the former case, which is the more general, shall be first considered, and then such modifications shall be introduced into the result as shall adapt it to the latter, and as the mutual interdependence of the variables may require.

For the sake of illustration let us consider a tree, and let us suppose its growth to depend on three circumstances which are independent of each other, viz. the fertility of the soil, the rain that waters it, and the heat of the sun: then, if the relation or law of connexion of these four things is expressed mathematically and in an explicit form, we shall have the growth of the tree a function of three variables; thus,

Growth of tree = F (fertile soil, rain, solar heat).

And if the law is known which connects the single effect with the three producing causes, then F, the symbol of the form of the function, is known.

Now observing the independence of the three variables in

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