Difference of the indices i and j in binary response models

I am currently trying to get the hang of binary probit and logit models as well as multinomial models, but I struggle to see the difference in the indices i and j respectively the difference in their meaning. For example, if we have the nonlinear model for probabilities

$$p_i = Pr(y_i = 1 \mid x_i) = G(x_i'\beta)$$

and calculate the level-dependent marginal effects (on the probability of success), we calculate

$$\frac{\delta E[y_i\mid x_i]}{\delta x_{ij}}=\frac{\delta Pr(y_i = 1 \mid x_i)}{\delta x_{ij}} = g(x_i'\beta)\beta_j$$

I am familiar with the index i referring to the i-th observation (of x or y), but I do not understand why the j-index is added to the x in the denominator and what it stands for and why it is added to the coefficient beta as well (at the end). Maybe someone can help, I'd appreciate it!

• In this setting $x_i$ is not a number but a vector $x_i = (x_{i1},\dots, x_{ip})$ and $x_{ij}$ denotes the $j$th element of the vector $x_i$. The same apply for $\beta = (\beta_1,\dots, \beta_p)$. In regression models, $x_i$ often represents a vector of covariates while $\beta$ is the vector of coefficients associated with those covariates. – winperikle Jan 21 at 13:27