Given $u(v,p)$ and $g(v,p)$ is relate by below equation
$$\frac{dp}{dv}=\frac{g-u_v}{u_p}=f(v,p)$$
This is how I got it but I do not know why a direct differentiation using$\frac{u}{v}$ method does not work. Unless I having the wrong concept regarding differential equation and taylor series expansion.
Consider equation below and $p(v_0)=p_0$
$u(p_1,v_0+h)-u(p_0,v_0)=g(p_0,v_0)h$
I use the same method as previous post Is the assumption consider cheating in mathematics? to find what is $p_1$ that satisfy the above equation with little changes, that is taking the coefficient of $h$ as $g(p_0,v_0)$ and coefficient for $h^n,n=2,3,4,...$ as $0$ and derive up to 3rd order expansion.
and letting
$f=\frac{g-u_v}{u_p}$
will get
$p_1=p_0+hf_0-\frac{h^2}{2}(\frac{u_{pp}f^2+2u_{pv}f+u_{vv}}{u_p})_0+O(h^3)$
since $u_pf+u_v=g$
$g_p=u_{pp}f+u_pf_p+u_{pv}$
$g_v=u_{pv}f+u_pf_v+u_{vv}$
$p_1=p_0+hf_0+\frac{h^2}{2}[f_pf+f_v-(\frac{g_pf+g_v}{u_p})]_0+O(h^3)$
I derive up to the 3rd oder and get
$p_1=p_0+hf_0+\frac{h^2}{2}[f_pf+f_v-(\frac{g_pf+g_v}{u_p})]_0+\frac{h^3}{6}[(f_pf+f_v)_pf+(f_pf+f_v)_v-(\frac{(g_pf+g_v)_pf+(g_pf+g_v)_v}{u_p})+\frac{3g_p(g_pf+g_v)}{u_p^2}-\frac{3f_p(g_pf+g_v)}{u_p}]_0+O(h^4)$
This method is very ineffective in deriving the term as for higher order term, it have more and more derivative to deal with, So I tried a direct differentiation on function of f as below.
First derivative
$f=\frac{g-u_p}{u_v}$
Second derivative
$f_pf+f_v$
$=(\frac{g-u_v}{u_p})_pf+(\frac{g-u_v}{u_p})_v$
$=\frac{u_pg_pf-u_pu_{pv}f-gu_{pp}f+u_vu_{pp}f+u_pg_v-u_pu_{vv}-gu_{pv}+u_vu_{pv}}{u_p^2}$
But it is different from the derivative from method above.
$f_pf+f_v-(\frac{g_pf+g_v}{u_p})$
Correction: I think i found my own mistake
Assuming $p_2=p_0+hf+\frac{h^2}{2}(f_pf+f_v)+...$
Inputting below equation and taking $g=u_pf+u_v$
$u(p_2,v_0+h)-u(p_0,v_0)$
$=gh+\frac{h^2}{2}(g_pf+g_v)+...$
Which mean a full taylor expansion of f will show full taylor expansion of g?
$u_p(\frac{h^2}{2}(f_pf+f_v)+\frac{1}{2}(u_{pp}(hf)^2+2u_{pv}(hf)(h)+u_vv(h^2))=\frac{h^2}{2}(g_pf+g_v)$
$f_pf+f_v=\frac{u_pg_pf-u_pu_{pv}f-gu_{pp}f+u_vu_{pp}f+u_pg_v-u_pu_{vv}-gu_{pv}+u_vu_{pv}}{u_p^2}$