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

CHANGE OF VARIABLE

95. Interchange of dependent and independent variables. 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, (§ 33), we have

(35)

${\frac {dy}{dx}}={\frac {1}{\frac {dx}{dy}}}$,

${\frac {dx}{dy}}\neq 0$

giving ${\tfrac {dy}{dx}}$ in terms of ${\tfrac {dx}{dy}}$. Also, by XXV,(§ 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

(A)	${\frac {d^{2}y}{dx^{2}}}={\frac {d}{dy}}\left({\frac {1}{\frac {dx}{dy}}}\right){\frac {dy}{dx}}$.

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

(36)

${\frac {d^{2}y}{dx^{2}}}=-{\frac {\frac {d^{2}x}{dy^{2}}}{\left({\frac {dx}{dy}}\right)^{3}}}$,