To obtain the formula for f'', you need to use the chain rule in a crafty way. Write $y=f(x)$. Then y satisfies the equation $F(x,y)=0$. Now regard F as a function of the two variables x and y, each of which is in turn a function of x (via the equations x=x and y=f(x)). Differentiate the equation $F(x,y)=0$ with respect to x, using the chain rule, and you get $F_x(x,y) + f'(x)F_y(x,y) = 0$. Now repeat the process, differentiating that equation with respect to x. This gives $F_{xx}(x,y) + f'(x)F_{xy}(x,y) + f''(x)\bigl(F_{yx}(x,y) + f'(x)F_{yy}(x,y)\bigr) = 0$.
Solve the first of those two equations to get an expression for f'(x). Substitute this into the second equation and you'll get the given expression for f''(x). (You'll need to quote the theorem which says that $F_{yx} = F_{xy}$ provided that these both exist and are continuous.)