Derivative of a constant matrix to the $t$th power with respect to the exponent $t$.

Question

Let $A$ be a constant matrix. Is there any useful formula for the derivative of $A^t$ with respect to $t$?

I know that for a matrix which can be represented as an exponential $\exp(B)$, we will have $\exp(B)^t = \exp(tB),$ and so $\frac d{dt} \exp(tB) = B\exp(tB)=B^{\frac 12}\exp(tB)B^{\frac12}.$

This suggests to me that for a general matrix $A$ the answer might be $\log(A)A$, where this makes sense. Is this correct?