- #1
gulsen
- 217
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partial derivate "philosophy"
I'm having problems with fundementals of partial derivatives. I've looked around for something like "partial derivative FAQ", but couldn't find.
Can any of the following be true in general?
[tex]\frac{\partial f}{\partial x} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}[/tex]
[tex]\frac{d f}{d x} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}[/tex]
I'd rather expect
[tex]\frac{d f}{d x} = \frac{d f}{d g}\frac{d g}{d x}[/tex]
but the partial version seems to be working equally. Can there be cases where the partial version fails?
Also, it's weird that although
[tex]\frac{d f}{d g} = \frac{1}{\frac{d g}{d f}}[/tex]
holds for all,
[tex]\frac{\partial f}{\partial g} = \frac{1}{\frac{\partial g}{\partial f}}[/tex]
doesn't. Why not?
I'm desperately looking for a proof for any of them...
p.s.: You may ask which one is independent variable and on what f and g depends. I leave it to the guy who's willing to prove them, because I'm not sure whether if makes any difference in a general proof.
I'm having problems with fundementals of partial derivatives. I've looked around for something like "partial derivative FAQ", but couldn't find.
Can any of the following be true in general?
[tex]\frac{\partial f}{\partial x} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}[/tex]
[tex]\frac{d f}{d x} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}[/tex]
I'd rather expect
[tex]\frac{d f}{d x} = \frac{d f}{d g}\frac{d g}{d x}[/tex]
but the partial version seems to be working equally. Can there be cases where the partial version fails?
Also, it's weird that although
[tex]\frac{d f}{d g} = \frac{1}{\frac{d g}{d f}}[/tex]
holds for all,
[tex]\frac{\partial f}{\partial g} = \frac{1}{\frac{\partial g}{\partial f}}[/tex]
doesn't. Why not?
I'm desperately looking for a proof for any of them...
p.s.: You may ask which one is independent variable and on what f and g depends. I leave it to the guy who's willing to prove them, because I'm not sure whether if makes any difference in a general proof.
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