We consider the following function

$$f:(\mathbb S^n)^N \rightarrow \mathbb R^{n+1} \text{ such that } f(x_1,...,x_N)= \sum_{i=1}^N x_i.$$

This function can be written in Cartesian coordinates as $f(x)=(f(x)_1,..,f(x)_{n+1})$ and I would like to know if one can find a simple expression for the derivative

$$\nabla_{x_1} \left(\frac{f(x)_1}{\Vert f(x) \Vert_{\mathbb R^{n+1}}}\right)$$ where $\nabla_{x_1}$ is the gradient on $\mathbb S^n$ with respect to $x_1.$

Can one somehow carry out this differentiation? I am a bit struggeling with computing $\nabla_{x_1} f(x)_1$ here.