Solve for $\frac{\partial u}{\partial x}$, where $f(x,y,u,v)$ using implicit function theorem

Question

I have pasted the problem and part of the solution below. This question is from Marsdens vector calculus in the section on the implicit function theorem. They did the first part by computing the determinant where each row of the matric contained in the component functions, and each column computed the partial derivative of that component function, first with respect to $u$ then with $v$.

My Question

To find $\dfrac{\partial u}{\partial x}$ why did they implicitly differentiate each component function at $x$, $u$, and $v$? Differentiating at $x$ then implictly differentiating $u$ with respect to $x$, is what I would've done, so I'm not sure why they also implicity differentiated $v$ with respect to $x$?

Thanks

Solution

Answer 1

The assumption is that both $u$ and $v$ are functions of $x$ and $y$

Here $x$ and $y$ are independent variables while $u$ and $v$ are dependent variables.

When you differentiate the equation involving both $u$ and $v$ with respect to $x$ we have to differentiate each and every term.

For example from $$ u+v=2x+y$$ we get $$\frac {\partial u}{\partial x} + \frac {\partial v}{\partial x} =2$$

Even though we are only interested in $\frac {\partial u}{\partial x}$ we still have to count $\frac {\partial v}{\partial x}$ in the equation.