Derivative of Multiplied Matrices

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I have the equation $P = A\cdot B^{-t/T}$ where:

  • $t$ is a variable scalar
  • $A$ is a matrix in terms of $t$
  • $B$ is a constant matrix
  • $T$ is a known constant scalar

I need to determine the derivative of $P$ in terms of $t$. I (probably mistakenly) assumed the product rule applied such that $\dfrac{dP}{dt} = A\cdot (-t B^{-t/T -1}) + (B^{-t/T})\cdot \dfrac{dA}{dt}$. but this does not provide the correct result when evaluated.

Is a solution even possible? Thanks in advance.