Confusion about matrix differentiation in a nonlinear matrix equation

I am trying to solve a matrix equation in the following discrete form:$$\frac{K^{n+1}-K^n}{\Delta t} = [(K^{n+1} (V^{n})^T).^3 - K^{n+1} (V^{n})^T]V^n.$$where $K^{n+1} \in \mathbb{R}^{m \times r}, V \in \mathbb{R}^{n \times r}$, $V^T V = I$(identity matrix) and $.^3$ are elementwise operations.

I am attempting to solve using Newton iterate method, which means I need to differentiate the function F：$$F = K^{n+1}-K^n-\Delta t[(K^{n+1} (V^{n})^T).^3 - K^{n+1} (V^{n})^T]V^n.$$I differentiate F with respect to $K^{n+1}$ and get:$$\frac{\partial F}{ \partial K} = I_K - \Delta t [3(K^{n+1} (V^{n})^T).^2 (V^{n})^T - (V^{n})^T]V^n.$$In my understanding, if a matrix is differentiated with respect to itself, the result should be an identity matrix. However, since $K \in \mathbb{R}^{m \times r}$ is not a square matrix, how should this be handled? How can I solve the matrix equation? Thanks in advance.