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

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

Dear all,

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

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

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

Any suggestions will be appreciated.

Thanks

2. Originally Posted by bugatti79
Dear all,

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

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

ok, I calculate

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

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

3. Originally Posted by bugatti79
ok, I calculate

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

I think $\displaystyle u_x_x$ 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

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

Thanks