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  4. (d2x/dy2) equals :

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Q. $\frac{d^{2}x}{dy^{2}} $ equals :

Continuity and Differentiability

Solution:

$\frac{d^{2}x}{dy^{2}} = \frac{d}{dy} \left(\frac{dx}{dy}\right) = \frac{d}{dx}\left(\frac{dx}{dy}\right) \frac{dx}{dy}$
$ = \frac{d}{dx}\left(\frac{1}{dy/dx}\right) \frac{dx}{dy} $
$ = - \frac{1}{\left(\frac{dy}{dx}\right)^{2}} . \frac{d^{2}y}{dx^{2}}. \frac{1}{\frac{dy}{dx}} $
$=- \frac{1}{\left(\frac{dy}{dx}\right)^{3}} \frac{d^{2}y}{dx^{2}}$