My question revolves around the following derivative:(adsbygoogle = window.adsbygoogle || []).push({});

for z(x,y)

*sorry I can't seem to get the latex right.

∂/∂x (∂z/∂y)

What I thought about doing was using the quotient rule to see what I would get (as if these were regular differentials). So, I "factored out" the 1/∂x, then did the quotient rule with the "differential," then divided by ∂x.

∂/∂x (∂z/∂y) = 1/∂x(∂ (∂z/∂y))

Doing the quotient rule with the bold:

1/∂x"(low d high - high d low )/low squared"

which gave:

1/∂x(∂y∂²z - ∂z∂²y)/(∂y²)

Now divide by ∂x:

(∂y∂²z -∂z∂²y)/(∂x∂y²)

Now, if I assume the bold above is somehow zero, suddenly I have the right answer:

(∂²z)/(∂x∂y)

Now, I know this is probably horrid math(I can't emphasize this enough. Battlemage! ≠ crank), but if only that second term in the top of the fraction is zero then it works.

So, my question is, is there any legitimacy whatsoever to this?

Oddly, if I do it with this:

∂/∂x (∂z/∂x)

I get again the same result, with the second term in the top of the fraction = 0, then it's the right answer:

1/∂x ("(low d high - high d low)/low squared" )

1/∂x ((∂x ∂²z - ∂z∂²x)/(∂x²) )

((∂x ∂²z - ∂z∂²x)/(∂x³)

assume right term in numerator = 0

(∂x ∂²z)/(∂x³) = ∂²z/(∂x²)

Just what is going on here...

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# Question about partial derivatives (it's probably based on flawed reasoning, but )

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