Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/175

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Therefore

(B) ;

and so on for higher derivatives. This transformation is called changing the independent variable from x to t. It is usually better to work out examples by the methods illustrated above rather than by using the formulas deduced.

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

(C)
(D) .
Solution.
  , therefore
(E) .
Also ; therefore
(F)
Also
Substituting in the last result from (E),
(G)
Substituting (D), (F), (G) in (C),
and reducing, we get
Ans.


Since the formulas deduced in the Differential Calculus generally involve derivatives of y with respect to x, such formulas as (A) and (B) are especially useful when the parametric equations of a curve are given. Such examples were given in §66, and many others will be employed in what follows.