How do you solve for Zx and Zy given Z(x,y)=x/y-y/x

Question

It's a Calculus 3 derivative based question.

Answer 1

I assume since you mentioned derivatives Zx and Zy are actually Z'x and Z'y (the partial derivatives).

 

Z'x can be thought of as Z'(x,y) where y is taken as a constant. So let's compute the derivative of Z(x,y) but treat y as a constant while we do that.

 

This gives us Z'x = d/dx(x/y - y/x) = 1/y + y/x^2

 

Step by step: The derivative of x/y, with y as a constant, is basically the derivative of some number (1/y) times x, so we just remove the x, leaving us with the constant (1/y). The derivative of -y/x is a little tricker; think of this as -y (the constant) times x^-1. Recall that the power rule has you first multiply by the power, so multiplying by -1 gives us -x^-1, and then we subtract one from the power giving us -x^-2, or -1/x^2. The two negatives cancel out and we are left with 1/y + y/x^2.

 

And lastly Z'y (taking x as a constant this time) = d/dy(x/y - y/x) = x/y^2 + 1/x

 

This one is basically the same process, just flipping x and y (treating x as constant and differentiating with respect to y).