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

 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.

Having given the equation

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

Solution. From (F),


 * Substituting in (E),


 * and reducing, we get $$\tfrac{d^2 z}{dx^2} - 2 \left( \tfrac{dz}{dx} \right)^2 = \cos^2 z$$. Ans.

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

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

By XXV, &sect;33,