TITLE: Derivative of matrix exponential w.r.t. to each element of the matrix
QUESTION [17 upvotes]: I have $x= \exp(At)$ where $A$ is a matrix. I would like to find derivative of $x$ with respect to each element of $A$. Could anyone help with this problem?

REPLY [3 votes]: According to Derivatives of the Matrix Exponential and Their Computation (who reference Karplus, Schwinger, Feynmann, Bellman and Snider) the derivative can be expressed as the linear map (i.e. Fréchet derivative)
$$\frac{\rm{d} e^{At}}{\rm{d} A} = \Big(V\longmapsto\int_{0}^t e^{A(t-\tau)}Ve^{A\tau}\,\rm{d}\tau\Big)$$