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Hi all, I need to understand these differential equations specially moving from the second order to the third order because i couldn't understand how they got to the result, what was exactly the principle:

$$ y'=f(x,y) $$

$$ y''=\frac{df}{dx}(x,y(x)) = f_{x}(x,y) + f_{y}(x,y)y' = f_{x}(x,y) + f_{y}(x,y)f(x,y) $$

$$ y'''=f_{xx}+2ff_{xy}+f_{yy}f^{2}+f_{x}f_{y}+ff_{y}^{2} $$

where $$ f_{x} $$ is the partial derivation of x and so for the similar other quantities.

please help me with it, thank you.

$$ y'=f(x,y) $$

$$ y''=\frac{df}{dx}(x,y(x)) = f_{x}(x,y) + f_{y}(x,y)y' = f_{x}(x,y) + f_{y}(x,y)f(x,y) $$

$$ y'''=f_{xx}+2ff_{xy}+f_{yy}f^{2}+f_{x}f_{y}+ff_{y}^{2} $$

where $$ f_{x} $$ is the partial derivation of x and so for the similar other quantities.

please help me with it, thank you.

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