# Finding Partial Derivative ($n$-dimensional) using implicit differentiation vs explicitly solving

This is a book example (not a homework question) about implicit differentiation on a composite of functions in $n$-dimensional space.

But my book explains this example in a very unclear manner. So I really appreciate your help in clarifications, since I'm very lost now >_<

Given:

$F(x,y,u,v) = x^2 + ux + y^2 + v$

$G(x,y,u,v) = x + yu + v^2 + x^2 v$

Goal: Find $\frac{\partial u}{\partial x}$ by 2 methods:

(1) by implicit differentiation

(2) explicitly solve for $u$

So far, I used the Jacobian to get the partial derivative, where:

$$\frac{\partial u}{\partial x} = \frac{-\left|\frac{\partial (F,G)}{\partial (x,v)}\right|}{\left|\frac{\partial (F,G)}{\partial (u,v)}\right|}$$

after some prep-calculations for partials of $F$ with respect to $x, u, v$; and partials of $G$ with respect to $x, u, v$

So my question is, is this method considered (1) or (2)? What does it really mean to "explicitly solve" for $u$? Just treat $x,y,v$ as constants and solve for $u$ in $F$ and in $G$?

My friend said if I do method (2), I shall get a different answer from my answer using Jacobian, but I don't know what to do to get the 2nd answer.

$F(x,y,u,v)=x^2+ux+y^2+v$,
$G(x,y,u,v)=x+yu+v^2+x^2v$.
Maybe for method (2), the idea is to eliminate $v$, then solve for $u$. So we can take $v=F-x^2-ux-y^2$, then substitute that into the second equations, expand/simplify, and try to solve for $x$. I suppose it is possible you will have a quartic equation in $x$, so if you don't see a trick for solving it, you may try to have some software solve that algebraic equation.