My Math Forum Nested partial derivatives, ('chain rule'?)

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 May 29th, 2009, 04:23 AM #1 Newbie   Joined: May 2009 Posts: 2 Thanks: 0 Nested partial derivatives, ('chain rule'?) Greetings. I'm slogging through some basic quantum mechanics at the moment, and as one could expect it is largely the level of difficulty of the mathematics that decides my progress rate. The last operation that has got me a bit stumped occurs when calculating the commutators for the angular momentum operators. Looking at the solution manual, a key bit of information is this: $$Y\frac{\partial }{{\partial z}}\left( {Z\frac{\partial }{{\partial x}}} \right)f = YZ\frac{{\partial ^2 f}}{{\partial z\partial x}} + Y\frac{{\partial f}}{{\partial x}}$$ Now, I can see how $$Y\frac{\partial }{{\partial z}}\left( {Z\frac{\partial }{{\partial x}}} \right)f = Y\frac{{\partial f}}{{\partial x}}$$ would make sense, the outermost operation is derivation with regard to Z, so that cancels the Z in the nested expression, right? I can also kind of agree with $$Y\frac{\partial }{{\partial z}}\left( {Z\frac{\partial }{{\partial x}}} \right)f = YZ\frac{{\partial ^2 f}}{{\partial z\partial x}}$$ Indeed, that would be my initial 'naive' interpretation. Now, what I can't get my head around is: How can it be both? Any light being shed on this matter would be greatly appreciated. Thanks in advance!
 May 29th, 2009, 04:30 AM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Nested partial derivatives, ('chain rule'?) This is actually a simple application of the product rule for differentiation: $Y\frac{\partial}{\partial Z}\left(Z\frac{\partial}{\partial X}\right)f=Y\frac{\partial}{\partial Z}\left(Z\frac{\partial f}{\partial X}\right)=Y\left[\left(\frac{\partial}{\partial Z}Z\right)\frac{\partial f}{\partial X}+Z\frac{\partial}{\partial Z}\frac{\partial f}{\partial X}\right].$
 May 29th, 2009, 04:49 AM #3 Newbie   Joined: May 2009 Posts: 2 Thanks: 0 Re: Nested partial derivatives, ('chain rule'?) The product rule, D'oh! *smacks forehead* Should have noticed that right away, but somehow several del operators together generate blinding death rays or something, preventing me from seeing the obvious... Thanks a lot! (Also for giving the quickest answer I ever got from the inernet. )

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### nested partial derivatives

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