Calculating 2nd Derivative of an implicit function using chain rule.

**bugatti79** · August 12th 2010, 12:55 PM

Dear all,

I get stuck when I try to determine $U_x_x$ where $u(x(\xi,\eta))$

I calculate $u_x=\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial u}{\partial \eta}\frac{\partial \eta}{\partial x}$

$u_x_x=\frac{\partial^{2} u}{\partial \xi^{2}}\frac{\partial^{2} \xi}{\partial x^{2}}$ ...... Something is telling me this is wrong but I dont know how to proceed any further

Any suggestions will be appreciated.

Thanks

**bugatti79** · August 14th 2010, 05:39 AM

Originally Posted by bugatti79

Dear all,

I get stuck when I try to determine $U_x_x$ where $u(x(\xi,\eta))$

I calculate $u_x=\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial u}{\partial \eta}\frac{\partial \eta}{\partial x})$

ok, I calculate

$u_x_x=\frac{\partial^{2}\xi}{\partial x^{2}}\frac{\partial u}{\partial \xi}}+\frac{\partial \xi}{\partial x}(\frac{\partial^{2} u}{\partial \xi^{2}}\frac{\partial \xi}{\partial x}+\frac{\partial^{2} u}{\partial \eta \partial \xi}\frac{\partial \eta}{\partial x})$

I think $u_x_x$ involves a chain rule within a product rule. Can anyone confirm this? Thanks

**bugatti79** · August 14th 2010, 05:45 AM

Originally Posted by bugatti79

ok, I calculate

$u_x_x=\frac{\partial^{2}\xi}{\partial x^{2}}\frac{\partial u}{\partial \xi}}+\frac{\partial \xi}{\partial x}(\frac{\partial^{2} u}{\partial \xi^{2}}\frac{\partial \xi}{\partial x}+\frac{\partial^{2} u}{\partial \eta \partial \xi}\frac{\partial \eta}{\partial x})$

I think $u_x_x$ involves a chain rule within a product rule. Can anyone confirm this? Thanks

Sorry I forgot to put in the second chunk, the complete equation is

$u_x_x=\frac{\partial^{2}\xi}{\partial x^{2}}\frac{\partial u}{\partial \xi}}+\frac{\partial \xi}{\partial x}(\frac{\partial^{2} u}{\partial \xi^{2}}\frac{\partial \xi}{\partial x}+\frac{\partial^{2} u}{\partial \eta \partial \xi}\frac{\partial \eta}{\partial x}) +\frac{\partial^{2}\eta}{\partial x^{2}}\frac{\partial u}{\partial \eta}}+\frac{\partial \eta}{\partial x}(\frac{\partial^{2} u}{\partial \eta^{2}}\frac{\partial \eta}{\partial x}+\frac{\partial^{2} u}{\partial \xi \partial \eta}\frac{\partial \xi}{\partial x})$

Thanks

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