I can derivate x(y) wrt y using the derivative of y(x) wrt x, follows the formula: [tex]\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}[/tex] until same the 2nd derivative (taking the 2nd diff form of x and deriving wrt to x):[tex]d^2x=\frac{d^2 x}{dy^2} dy^2 + \frac{dx}{dy} d^2y[/tex] [tex]\frac{d^2x}{dx^2}=\frac{d^2 x}{dy^2} \frac{dy^2}{dx^2} + \frac{dx}{dy} \frac{d^2y}{dx^2}[/tex] [tex]0=\frac{d^2 x}{dy^2} \frac{dy^2}{dx^2} + \frac{dx}{dy} \frac{d^2y}{dx^2}[/tex] solving for d²x/dy²: [tex]\frac{d^2x}{dy^2}=-\frac{d^2y}{dx^2}\frac{1}{\left( \frac{dy}{dx} \right)^3}[/tex] I think that this is a razoable deduction for the formula of d²x/dy² in terms of derivatives of y wrt x.(adsbygoogle = window.adsbygoogle || []).push({});

Now, where it came from this formula???

links:

http://en.wikipedia.org/wiki/Integration_of_inverse_functions

http://en.wikipedia.org/wiki/Inverse_functions_and_differentiation

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# Integral by inverse function

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