I need your help with the differential of a matrix exponential.
A function $f$ is a mapping from a matrix to a matrix, that is, $f: \mathbb{R}^{n \times n}\rightarrow \mathbb{R}^{n \times n}$. Here $\mathbf{H} \in\mathbb{R}^{n \times n} $ is a matrix. And each element of $\mathbf{H}$ is parameterized by parameter $\theta$.
In this situation, I would like to know if it is correct $$ \frac{d e^{f(\mathbf{H})}}{d \theta} = \frac{d f(\mathbf{H})}{d\theta}e^{f(\mathbf{H})} $$
According to this document, it holds that $$ \frac{d}{dt}\exp{\left( \mathbf{H}(t) \right)} = \exp{\left( \mathbf{H} \right)} \left\{ \mathbf{H}'-\frac{1}{2!}[\mathbf{H},\mathbf{H}']+\frac{1}{3!}[\mathbf{H},[\mathbf{H},\mathbf{H}']]-\frac{1}{4!}[\mathbf{H},[\mathbf{H},[\mathbf{H},\mathbf{H}']]]+\cdots \right\} $$
where $[\mathbf{A},\mathbf{B}] = \mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A}$ and
$$ \mathbf{H}'=\frac{d}{dt}\mathbf{H}. $$
So, the answer of the question is no.