I need help with a question.
Let $A$ and $B$ be real or complex-valued matrices. Define $[A,B] = BA-AB$. Prove that if $[A,[A,B]]=[B,[A,B]]=0$ then, for every $t \in \mathbb{R}$:
$e^{tB}e^{tA}=e^{t(A+B)}e^{\frac{t^2}{2}[A,B]}$
Hint: Verify that $\Phi(t)=e^{-t(A+B)}e^{tA}e^{tB}$ is a solution of
$X'=t[A,B]X$.
I know how to prove that if $\Phi$ is a solution of the ODE then the identity holds. But I think I am miscalculating the derivative of $\Phi$. I can't see how that $t$ would appear in the expression of the derivative.