Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/173

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giving in terms of and . Similarly,

(37) ;  

and so on for higher derivatives. This transformation is called changing the independent variable from x to y.

Illustrative Example 1. Change the independent variable from x to y in the equation

Solution. Substituting from (35), (36), (37),

Reducing, we get

a much simpler equation.

96. Change of the dependent variable. Let

(A)

and suppose at the same time y is a function of z, say

(B)

We may then express , etc., in terms of , , etc., as follows

In general, z is a function of y by (B), §42; and since y is a function of x by (A), it is evident that z is a function of x. Hence by XXV we have

(C) .
Also . By V, §33
But By XXV, §33
(D)