What Are the Dimensions for Force and Distance in This Problem?

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The discussion focuses on determining the dimensions of force (P) and distance (x, y, z) in a physics problem. The dimensions for force are established as (ML/T²), where M represents mass, L represents length, and T represents time. The user inquires about the dimensions of the derivative dP/dx, concluding that it is M/T², and questions whether the second derivative retains the same dimensions despite differentiating with respect to different variables. The conversation also touches on the relationship between derivatives and their dimensional ratios.

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Homework Statement


Let P represent a force and x, y, and z represent distances. Determine the dimensions for each of the quantities listed below.

I attached the problem I need help with. (it's a small picture)
So I'm a bit confused since the it involves both dx and dy.

The dimensions for P is (ML/T^2)
Where M=mass
L=length
T=time

Then I have to differentiate it twice but with two different variables. Since x and y refer to distances and the only distance found in that equation is L.

Would that mean the dimensions for dP/dx = M/T^2 ?

If so, would that mean the answer to the 2nd derivative still be the same answer since they're differentiating between different variables? (I'm not sure if I worded this right, but I remember from Calc III something similar)
On a side note. I took calculus quite a while ago so some of this I might have forgotten.
The picture I provided, that is not equivalent to (dP/dx)*(dP/dy) right?
 

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A derivative is the limit of a certain ratio. It has the same dimensions as the ratio. What dimensions does the ratio have?
 

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