Consider the following expressions:
$$C_{i}=\sum_{j=1}^{N_i}v_{j}, \quad v_j \in \mathbb{R}, \quad N_i \in \mathbb{N} $$
\begin{equation} x_{i}=\frac{C_{i}}{N_i} \end{equation}
I want to obtain an expression for $\frac{\partial x_i}{\partial C_{i}}$. My first approach was to simply use the second definition hence I obtained:
$$\frac{\partial x_i}{\partial C_{i}}=\frac{1}{N_i}$$
However, someone suggested that this was somehow wrong because $C_{i}$ was a function and not a variable. So he told me to recurr to the implicit function theorem. Listening to his suggestion, I tried using the chain rule:
$$\frac{\partial x_i}{\partial C_{i}}=\frac{\partial x_i}{\partial v_1}\frac{\partial v_1}{\partial C_{i}}+...+\frac{\partial x_i}{\partial v_{n_i}}\frac{\partial v_{n_i}}{\partial C_{i}}=\frac{1}{N_i}N_i=1$$
Which one do you think is correct? And why? I am sure mine, but I want an argument to prove this person wrong.