How to take derivative of Ky Fan $k$ norms of a matrix $A$: $\frac{\partial\|A\|_k}{\partial A}$

Question

The Ky Fan $k$ norms of matrix $A$ is define as

$$ \|A\|_k = \sum_{i=1}^k \sigma_i(A)$$

where $ \sigma_i(A)$ is the singular value of $A$ in descent order.

Is it possible to compute the Ky Fan $k$ norms of matrix $A$ since the Ky Fan $k$ norms is convex:

$$\frac{\partial\|A\|_k}{\partial A}$$

Note: $A$ can be nonnegative.