Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/265

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Evidently (B) is the value of (C) when . Considering (C) as a function of, we may write

(D)

which may then be expanded in powers of by Maclaurin's Theorem, (64), § 145, giving

(E)

Let us now express the successive derivatives of with respect to in terms of the partial derivatives of with respect to and . Let

(F)

then by (51), §125,

(G)

But from (F),

(H) and

and since is a function of and through and ,

and

or, since from (F), and ,

(I) and

Substituting in (G) from (I) and (H),

(J)

Replacing by in (J), we get

(K)

In the same way the third derivative is

(L)

and so on for higher derivatives.