Surfaces $x^2+y^2+z^2+w^2=1$ and $x+2y+3z+4w=2$ define implicitly functions $x=x(y,z)$ and $w=w(y,z)$. Show that $\frac{\delta x}{\delta y}=\frac{2x-4y}{4x-w}$ when $4x \neq w$

Attempt to solve

Now i am not quite sure what i am suppose to do here ? I suppose compute $\frac{\delta x}{\delta y}$ via implicit-differentiation since both of these surfaces are given in implicit form. Another problem is that i am suppose to form system of equations from two of these surfaces since $x=x(y,z)$ and $w=w(y,z)$ are referring to same function ? Something like this ?

$$ F(x(y,z),y,z,w(y,z))=\begin{cases} x^2+y^2+z^2+w^2=1 \\ x+2y+3z+4w=2 \end{cases} $$

I don't think i have very good understanding on what i am actually suppose to do here so if someone can hint me in right direction / maybe try to explain on what i am suppose to do. I don't expect someone to handover complete solutions.

up vote 1 down vote accepted


(1) Write $x,w$ as functions and the other variables as... variables: $$x(y,z)^2 + y^2 + z^2 + w(y,z)^2 = 1,$$ $$x(y,z) + 2y + 3z + 4w(y,z) = 2.$$ (2) Apply $\partial_y$,... to each equation: $$2x(y,z)x_y(y,x) + 2y + 2w(y,z)w_y(y,x) = 0,$$ $$\cdots$$ (3) Solve the (linear) system with unknowns $x_y,w_y,x_z,w_z$.

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