Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/172

 CHANGE OF VARIABLE

It is sometimes desirable to transform an expression involving derivatives of y with respect to x into an equivalent expression involving instead derivatives of x with respect to y. Our examples will show that in many cases such a change transforms the given expression into a much simpler one. Or perhaps x is given as an explicit function of y in a problem, and it is found more convenient to use a formula involving $$\tfrac{dx}{dy}$$, $$\tfrac{d^2 x}{dy^2}$$, etc., than one involving $$\tfrac{dy}{dx}$$, $$\tfrac{d^2 y}{dx^2}$$, etc. We shall now proceed to find the formulas necessary for making such transformations.

Given $$y =f(x)$$, then from XXVI, (&sect; 33), we have

giving $$\tfrac{dy}{dx}$$ in terms of $$\tfrac{dx}{dy}$$. Also, by XXV,(&sect; 33),

$$\frac{d^2 y}{dx^2} = \frac{d}{dx} \left( \frac{dy}{dx} \right) = \frac{dy}{dy} \left( \frac{dy}{dx} \right) \frac{dy}{dx}$$

or

But $$\frac{d}{dy} \left( \frac{1}{\frac{dx}{dy}} \right) = - \frac{\frac{d^2 x}{dy^2}}{\left( \frac{dx}{dy} \right)^2}$$; and $$\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$$ from (35).

Substituting these in (A), we get