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Given two functions $f=f(x)$ and $u=u(x,y,z)$, where $x,y,z$ are independent, how do I get the second order derivative $\partial^2f/\partial u^2$?

My attempt:

$$\frac{\partial^2 f}{\partial u^2}=\frac{\partial}{\partial u}\frac{\partial f}{\partial u}=\frac{\partial}{\partial u}\left[ \frac{df}{dx} \left( \frac{\partial u}{\partial x} \right)^{-1} \right]$$

Then I can't proceed. I remember that there were plenty of this sort of questions in the Multivariable Calculus class, but clearly I thought them too easy to go into the note.

arax
  • 2,779
  • one thing is when $f$ depends on $u$, and then $u$ depends on $x$, so there's a chain rule. This case is not the same (both $f$ and $u$ depend on $x$), so you probably have to use some kind of inverse fuction theorem application to find $x_u$ – cjferes Oct 07 '14 at 15:57

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