A particle moves on a surface find ∂x/∂y at given point

[MeMoses]

Homework Statement

A particle moves on the surface (z^2)/2+yz-(x^2)/2=2. At point (-2, 1, 2) x is changing at the rate of 2/m sec and z is changing at the rate of -1/m sec. Determine in m/sec. the rate of change (with respect to time) of ∂x/∂z at (-2, 1, 2).

Homework Equations

The Attempt at a Solution

Not really sure where to start and all my attempts get me nowhere. Does ∂x/∂t = 2 and ∂z/∂t = -1? I tried that and then differentiated with respect to t and got nowhere. If someone could help me in the right direction that would be great. Thanks for your time.

 

[HallsofIvy]

Frankly, the phrase "the rate of change (with respect to time) of ∂x/∂z" doesn't make much sense to me. I guess it would mean
$$\frac{d\frac{\left(\partial x\right)}{\partial z}}{dt}$$
but that looks very peculiar.

An obvious problem would be to find the rate of change of y with respect to t.

 

[MeMoses]

That's what confuses me a bit. How can you find ∂x/∂z though?

 

[MeMoses]

Ok can someone correct me if this idea is completely wrong. Can ∂x/∂z be found (if the surface is F) by taking (∂F/∂z)/(∂F/∂x)? If that is allowed though I get (z+y)/x and when I differentiate with respect to time I still get a ∂y/∂t which I am not given a value for.