Derivative of $\theta=(\bar{X},X_i)$ with respect to $X_i$?

Question

Given a sample of random variables $X_1,\ldots,X_n$ and the vector $\theta=(\bar{X},X_i)$? What is the derivative of $\theta=(\bar{X},X_i)$ with respect to $X_i$ and with respect to $\bar{X}$? where $\bar{X}$ is the average of the sample. I'm particularly confused about $\dfrac {\delta \theta}{\delta X_i} $, is it $(0,1)$ (we consider $\bar{X}$ as a new variable $Y$)? or is it $(1,1)$ (we that $\bar{X}=\dfrac{1}{n}\sum X_i$ is a function of $X_i$)?