Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/174

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Similarly for higher derivatives. This transformation is called changing the dependent variable from y to z, the independent variable remaining x throughout. We will now illustrate this process by means of an example.

Illustrative Example 1. Having given the equation

(E)

change the dependent variable from y to z by means of the relation

(F)

Solution. From (F),

,

Substituting in (E),

and reducing, we get . Ans.

97. Change of the independent variable. Let y be a function of x, and at the same time let x (and hence also y) be a function of a new variable t. It is required to express

, etc.,

in terms of new derivatives having t as the independent variable.

By XXV, §33,

  , or
(A) .
Also
But differentiating (A) with respect to t,