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## Main Question or Discussion Point

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

[itex]f(u,v) = 0[/itex]

[itex]u = lx + my + nz[/itex]

[itex] v = x^{2} + y^{2} + z^{2}[/itex]

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

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

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

[itex] \frac{∂z}{∂x} = -\frac{\frac{∂(u,v)}{∂(x,y)}}{\frac{∂(u,v)}{∂(z,y)}}[/itex]

[itex] \frac{∂z}{∂y} = -\frac{\frac{∂(u,v)}{∂(y,x)}}{\frac{∂(u,v)}{∂(z,x)}}[/itex]

[itex]f(u,v) = 0[/itex]

[itex]u = lx + my + nz[/itex]

[itex] v = x^{2} + y^{2} + z^{2}[/itex]

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

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

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

[itex] \frac{∂z}{∂x} = -\frac{\frac{∂(u,v)}{∂(x,y)}}{\frac{∂(u,v)}{∂(z,y)}}[/itex]

[itex] \frac{∂z}{∂y} = -\frac{\frac{∂(u,v)}{∂(y,x)}}{\frac{∂(u,v)}{∂(z,x)}}[/itex]

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