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In "Introduction to Calculus and Analysis" pages 221-223 Courant derives the following for an implicit function F(x,y)=0. Using

$dF = F_x dx + F_y dy = 0$

$dy = \frac{dy}{dx} dx = -\frac{F_x}{F_y}dx$

Also $y' = -\frac{F_x}{F_y}$

He says $f(x) = y$ therefore

$y' = \frac{F_x(x,f(x))}{F_y(x,f(x))}$

What I don't understand is, when I derived $y''$ using the Quotient Rule

$y'' = -\frac{F_yF_{xx}f' - F_xF_{yx}f'}{F_y^2}$

in the book however the result is

$y'' = - \frac{F_yF_{xx}+F_yF_{xy}f' - F_x F_{xy} - F_x F_{yy}f'}{F_y^2}$

I dont understand how the extra terms were derived.

Thanks,

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1 Answer 1

up vote 1 down vote accepted

It is just the Chain Rule. When deriving the numerator with respect to $x$ we get: $$ \frac{\partial}{\partial x}F_x(x,f(x))=\frac{\partial F_x}{\partial x}(x,f(x))+\frac{\partial F_x}{\partial y}\frac{df}{dx}=F_{xx}(x,f(x))+F_{xy}(x,f(x))f'(x). $$ Similarly, $$ \frac{\partial}{\partial x}F_y(x,f(x))=F_{xy}(x,f(x))+F_{yy}(x,f(x))f'(x). $$

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