How do you get substitution to recognized a derivative?

I have a simple expression involving a derivative:

x[t_] = Q/T[t]; Simplify[Derivative[1][x][t] /. Q -> x T[t]] 

That produces:$ $ -\frac{x T'[t]}{T[t]}$ $ I have another relation: $ T=a^{-1}$ . When I make the substitution, I get this:

FullSimplify[-((x*Derivative[1][T][t])/T[t]) /. T[t] -> 1/a[t]] 

$ $ -x a[t] T'[t]$ $

But what I want is this: $ $ -\frac{x a[t]}{a'[t]}$ $ I know I can force the issue with an additional substitution of $ T'[t]=a'[t]^{-1}$ , but it seems to me that I’ve already given Mathematica everything it needs to make that substitution on it’s own. How do I force MMa to recognize this kind of substitution in the derivative?