# First-order ODEs, matrix exponentials, and det(exp)

Last time we derived a formula for the derivative of the matrix exponential. Here we will be focusing instead on the expression $$D\exp(x)u=\exp(x)u=u\exp(x),$$ which holds whenever $$u$$ commutes with $$x$$. In this post, $$E$$ denotes a real Banach space and $$L(E)$$ denotes the space of continuous linear operators on $$E$$. ODEs The (matrix) exponential can […]