- #1
Battlemage!
- 294
- 45
My question revolves around the following derivative:
*sorry I can't seem to get the latex right.
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.
Doing the quotient rule with the bold:
which gave:
Now divide by ∂x:
Now, if I assume the bold above is somehow zero, suddenly I have the right answer:
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...
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...