First prolongation formula

My question is about the derivation of the prolongation formula from Olver’s book:"Applications of Lie groups to differential equations" Page 109.

Considering a differential equation with independent variable(x) and one dependent variable(u):
(x,u) $subset$ $X times U$

The coordinate in first jet space $M^{(1)}$ is (x,$u^{(1)}$) = (x,u,$u_j$).

Let u = f(x) is any function with $u_j = frac{partial u}{partial x_j} $

First prolongation of a group action on M is given as:
$pr^{1} g_epsilon . (x,u^{(1)}) = (tilde{x},tilde{u}^{(1)})$

The dependent variable is unchanged here; I mean

$tilde{u} = tilde{f}_epsilon(tilde{x}) = f[Xi^{-1}_epsilon(tilde{x})] = f[Xi_{-epsilon} (tilde{x})] $

To find the infinitesimal generator of pr(1) go we must differentiate the
formulas for the prolonged transformations with respect to e and set e = O.
Thus
$pr ^{1} v= xi ^{i}(x)frac{partial}{partial x^i}+ phi ^{j}(x,u^{1})frac{partial}{partial u^j},$
where
$ phi ^{j}(x,u^{1})= frac{d}{ depsilon}_{epsilon|=0} [frac{partial Xi^k_{-epsilon}}{partialtilde{x}^j} ] (Xi_{epsilon}(x)). u_{k}= – frac{partial xi^k}{partial x^j}(x). $

It seems simple, but I don’t know how to calculate $ phi ^{j}(x,u^{1}) $. Can anybody help me to find the answer?
What kind of derivative is it?

Olver obtained two types of terms multiplying $u_{k}$, first:

$frac{partial}{partialtilde{x}^j} [frac{dXi^k_{-epsilon}}{depsilon} ] (Xi_{epsilon}(x)) |_{epsilon = 0} = frac{partial}{partial x^j} [frac{dXi^k_{epsilon}}{depsilon} ] |_{epsilon = 0} = – frac{partial xi^k}{partial x^j}(x)$

and second,

$ frac{partial^{2} [Xi^k_{-epsilon} ]}{partialtilde{x}^j partialtilde{x}^l} ((Xi_{-epsilon}(x)) [frac{dXi^k_{epsilon}}{depsilon} ](x) |_{epsilon = 0} =0.$

Shoud I add this two term? How can I obtained them?
Thank you in advance!