I'm having trouble understanding where this concept comes from:(adsbygoogle = window.adsbygoogle || []).push({});

Step 1) If you start out with the following two equations

v + log u = xy

u + log v = x - y.

Step 2) And then perform implicit differentiation, taking v and u to be dependent upon both x and y:

(d will represent the partial derivative symbol):

dv/dx + (1/u)du/dx = y

du/dx + (1/v)dv/dx = 1

I can do some simple Gaussian reduction and obtain:

du/dx = [u(v-y)]/[uv-1] which is the same answer my book gives, the only difference is that my book uses this method to find du/dx:

Step 3)

det(Row 1:yu, u; Row 2: v, 1)/det(Row 1: 1, u; Row 2: v, 1)

which reduces to: (yu)(1) - (u)(v))/(1)(1) - (u)(v).

If the two matrices were A, and B, respectively, then:

a11 = yu, a12 = u, a21 = v, a22 = 1.

b11 = 1, b12 = u, b21 = v, b22 = 1

(sorry, I don't know how to put a real matrix into here)

And my problem is that I just don't understand where these matrices came from. I think this may be some formula from linear algebra that I just don't remember, but my book gives no reference to what it's doing, and goes directly from step 2 to step 3, so I'm really kind of lost right now. Any help would be appreciated.

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# Matrix determinants and differentiation

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