Define a linear operator
$$ LP=A^TP+PA, $$ where $A:\mathbb R\rightarrow \mathbb R^{m\times m}$ and $P:\mathbb R\rightarrow \mathbb S^{m\times m}$
Define a differential operator $\partial_ k=\partial /\partial k$ where $k\in\mathbb R$, i.e. $\partial_k A=\frac{\partial A(k)}{\partial k}$
Is the following equation correct?
$$ \partial_k(LP)=(\partial_k L)P +L(\partial_k P) $$
If it is correct, the question is what is the meaning of $(\partial_k L)P$. $L$ is an operator itself but why we can put a linear operator $L$ to a differential operator $\partial_k$. The last part $L(\partial_k P)$ means a function $P$ after a differential operator $\partial_k$ and then after a linear operator $L$. What does $(\partial_k L)P$ mean?
New edit
I notice if $(\partial_k L)P$ is defined as
$$ (\partial_k L)P = (\partial_k A^T)P+P(\partial_k A), $$
the above equation is correct. But I can't find a solid reason why it should be defined like this.
I don’t think this directly answers the question, but I’m leaving it as a long comment.
One approach here to compute the partial derivative is as follows: for $v \in \Bbb R^n$, $$ \partial_k(LP)(v) = \lim_{t \to 0} \frac{LP(v + tk) - LP(v)}{t} = \\ \lim_{t \to 0} \frac{A^T(v+tk)P(v + tk) + P(v + tk)A(v+tk) - A^T(v)P(v) + P(v)A(v)}{t} = \\ \lim_{t \to 0} \frac{(A(v) + t\partial_k A)^T (P(v) + t\partial_k P(v)) + (P(v) + t\partial_k P(v))(A(v) + t\partial_kA) - A^T(v)P(v) + P(v)A(v)}{t} = \\ \lim_{t \to 0} \frac{t[(\partial_k A(v))^TP(v) + A(v) \partial_kP(v) + P(v) \partial_k A(v) + \partial_k P(v) A(v)] + o(t)}{t} =\\ (\partial_k A(v))^TP(v) + A(v) \partial_kP(v) + P(v) \partial_k A(v) + \partial_k P(v) A(v) $$