I have never been good at differentiation of implicit functions in cases when in a function is given, much less in abstract cases with composite functions. Hopefully someone can help me get started on the process to answering this homework question:
Suppose $y = f(x)$ is a solution of $G(x,y) = 0$, where $\frac{\partial{G}}{\partial{y}} \neq 0$, and let $g(x)=F(x, f(x))$. Show that $g'(x) = \frac{(F_1 G_2 - F_2 G_1)}{G_2}$, the right side being evaluated with $y = f(x)$.
My tutor and I have attacked this from every angle and can't figure it out. Please help!
Since $G(x,f(x))=0$
$$dG=\frac{\partial G}{\partial x}dx+\frac{\partial G}{\partial f}\frac{d f}{d x}dx=0$$ $$\Rightarrow \frac{d f}{d x}=-\frac{\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial f}}$$ and $$dg=dF(x,f(x))=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial f}\frac{d f}{d x}dx$$ $$\frac{dg}{dx}=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial f}\frac{d f}{d x}$$ by replacing $\frac{d f}{d x}$ $$\frac{dg}{dx}=\frac{\partial F}{\partial x}-\frac{\partial F}{\partial f}\frac{\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial f}}$$ $$\frac{dg}{dx}=\frac{\frac{\partial F}{\partial x}\frac{\partial G}{\partial f}-\frac{\partial F}{\partial f}\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial f}}$$