Implicit partial differentiation functional det[]

[54stickers]

Hello! I was wondering how I could find the following derivatives from the given function using Jacobian determinants.

$f(u,v) = 0$

$u = lx + my + nz$

$v = x^{2} + y^{2} + z^{2}$

$\frac{∂z}{∂x} = ?$ (I believe y is constant, but the problem does not specify)

$\frac{∂z}{∂y} = ?$ (I believe x is constant, but the problem does not specify)

I tried setting up the jacobian this way (failed):

$\frac{∂z}{∂x} = -\frac{\frac{∂(u,v)}{∂(x,y)}}{\frac{∂(u,v)}{∂(z,y)}}$

$\frac{∂z}{∂y} = -\frac{\frac{∂(u,v)}{∂(y,x)}}{\frac{∂(u,v)}{∂(z,x)}}$

 

[chiro]

Hello 54stickers and welcome to the forums.

Are you familiar with the chain rule and implicit differentiation? It would probably be a lot better for your understanding if you used these from first principles.

If you are given some identities to work with that are more developed than having to use first principles, you should also point these out on your end.

 

[54stickers]

I am familiar with chain rule and implicit differentiation, but I don't know how to go about that approach either.

also, what do you mean by 'first principles'?

 

[chiro]

Pretty much in this context, first principles means the chain rule and other similar rules (like implicit differentiation). I don't mean deriving everything from limits (which is one possible interpretation of first principles and I think its a good thing to clear up any misconceptions).

If you know these basic rules, you should be able to derive any necessary results that are required, even if they relate to multivariable problems (including partial derivatives).