$MRS = \frac{u_1}{u_2}$
Differentiating with respect to $x_1$:
the book shows that you get $$\frac{u_2\bigl( u_{11}+u_{12}\frac{dx_2}{dx_1}\bigr) - u_1\bigl( u_{21}+u_{22}\frac{dx_2}{dx_1}\bigr)}{u_2^2}.$$
I didn't understand how the book found this?
Here, $u_{11}$ is the second order partial derivative of the utility function with respect to $x_1$
This is the logic behind the notation of other partial derivatives. This is so since there are two goods: $x_1$ and $x_2$. The economic logic is that when we are accounting changes in the utility function with respect to good 1, there are changes in good 2 which also needs to be taken into account.
To give some more context, this is Marginal rate of substitution in economics and we are doing the second order condition.
$u_1 is the Marginal utility of good 1. Similarly for the other one. In economics, marginal utility is found by differentiating utility.
The first equation in the quetion (MRS) is found by:
$u_1dx_1 + u_2dx_2 = 0$
From here MRS = $dx_2/dx_1 = [u_1/u_2]$
(This is in absolute value form)
(edit: I answered before I see your edit on the question, so my answer was about how we derive the MRS).
The marginal rate of substitution is the slope of the indifference curve which represents the utility gained from bundles of, usually, two goods, $$u(x_1, x_2)$$ As usually the two goods are substitutes; to get one more unit from one good you have to give up "some" units of the other good. In other words, MRS is the ratio between the marginal utility of both goods. $$MRS= -{MU_1\over MU_2}$$
The marginal utility of good 1 is the change in the utility with respect to $\text{$x_1$}$: $$MU_1 = \frac{\Delta u}{\Delta x_1} = \frac{u (x_1 + \Delta x_1, x_2) - u(x_1,x_2)}{\Delta x_1}$$ $$MU_1 = {∂ \, u(x_1,x_2)\over ∂ \, x_1}$$ And you can do the same for good 2; the partial derivative with respect to $\text{$x_2$}$: $$MU_2 = {\partial \, u(x_1,x_2)\over \partial \, x_2}$$
Or, we can consider the indifference curve as a relationship so that we can express the value of good 2 as a function of good 1: $$x_2= x_2(x_1)$$ $$∴ u (x_1, x_2(x_1))=a \quad {\{a \in \Bbb R \, \vert \, a \gt 0\}}$$
Differentiate it with respect to $\text{$x_1$}$: $${\partial u(x_1,x_2(x_1))\over \partial x_1} + {\partial u(x_1,x_2(x_1))\over \partial x_2}{dx_2(x_1)\over dx_1}=0$$ $${dx_2(x_1)\over dx_1} = - {{\partial u(x_1,x_2(x_1))\over \partial x_1}\over {\partial u(x_1,x_2(x_1))\over \partial x_2}} = -{MU_1\over MU_2}$$