f(x_{1},x_{2},x_{3})= x^{2}_{1}e^{(}^{3x2+x1x3)}+(2x^{3}_{2}/x_{1})

Show that f12=f21 and f13=f31, which are implications of Young's Theorem.

What I did was FOCs with regards to x1,x2,x3. So I ended up with;

df/dx1= 2x1e^{(3x2+x1x3) }-2x_{2}^3/x_{1}^2

df/dx2= 3x^{2}e^{(3x2+x1x3) }+6x_{2}^2/x_{1 }df/dx3=x^{3}_{1}e^{(3x2+x1x3)}

....After many attempts I was unsuccessful at cracking the code. The solution guide states:

f12=f21= (6x_{1}+3x^{2}_{1}x_{3})e^{(3x2+x1x3)}-6x^{2}_{2}/x^{2}_{1 }f13=f31= (3x^{2}_{1}+x^{3}_{1}x_{3})e^{(3x2+x1x3)}

If anyone could fully explain this to me, I would really appreciate it because I am very perplexed as to how they logically came up with the answer. Thank you very much.