Suppose there's square matrix $\hat{A}$, whose entrances are differentiable functions $f_{ij}(x_1, x_2,..., x_n)$. With $\mu = 1, ..., n$. I want to calculate such formula
$$\frac{\partial}{\partial x_{\mu}} \exp{(A)}$$
Using series expansion for matrix exponential I derived following relation
$$\frac{\partial}{\partial x_{\mu}} \exp{(A)} = \exp(\frac{\partial}{\partial x_{\mu}} A) - \hat{1}$$,
where $\hat{1}$ is $n \times n$ square identity matrix. Since I haven't seen this formula in any textbook or script I just wanted to ask if it's true for any matrix exponential.