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,