Extremely hard for me at least!

This problem is just something I came up with. for practicing properties, and how to use partial.

I just want to know if I did this right.. Sorry, but no one I know can do this stuff, not even the cal teacher at my higschool .

consider: $\displaystyle t=x^{x^y}$

Find $\displaystyle \frac{\partial t_x}{\partial x}$ and

Then, say $\displaystyle \frac{\partial t_x}{\partial x}=z_x$

Find $\displaystyle \nabla z_x$

I found (hopefully correctly) that $\displaystyle z_x=x^{x^y}\left [ yx^{y-1} \right ] ln(x)+x^{y-1}$

Then

$\displaystyle \nabla z=\frac{\partial z_x}{\partial x}\mathbf{i}+\frac{\partial z_y}{\partial y}\mathbf{j}}$

$\displaystyle z=\left [x^{x^y} y\right]\left [x^{y-1} ln(x)\right ]+x^{y-1}$

$\displaystyle \frac{\partial z}{\partial x}=\frac{\partial}{\partial x}\left [x^{x^y} y\right]\left [x^{y-1} ln(x)\right ]+\left [x^{x^y} y\right]\frac{\partial}{\partial x}\left [x^{y-1} ln(x)\right ]+(y-1)x^{y-2}$

$\displaystyle \frac{\partial z}{\partial x}=\left [ {yz_x} \right ]\left [ x^{y-1}ln(x) \right ]+ \left [ \left [ (y-1)x^{y-2}ln(x) \right ]+x^{y-2} \right ]yx^{x^y}+(y-1)x^{y-2}$

$\displaystyle =\left [y \left [x^{x^y}\left [ yx^{y-1} \right ] ln(x)+x^{y-1} \right] \right ]\left [ x^{y-1}ln(x) \right ] + \left [ \left [ (y-1)x^{y-2}ln(x) \right ]+x^{y-2} \right ] yx^{x^y}+(y-1)x^{y-2}$

$\displaystyle \frac{\partial z}{\partial x}\mathbf{i}=\left [\left [\left [x^{x^y}\left [ yx^{y-1} \right ] ln(x^y)+yx^{y-1} \right] \right ]\left [ x^{y-1}ln(x) \right ] + \left [ \left [ (y-1)x^{y-2}ln(x) \right ]+x^{y-2} \right ] yx^{x^y}+(y-1)x^{y-2}\right ] \mathbf{i}$

Is that right?

I can't figure out the partial with respect to y though.. Can someone help?