How can we find the partial derivative of $\displaystyle x^{t-1}$ with respect to t? This looks as if it should be really easy, but I'm not seeing it...
Treat $\displaystyle x$ as a constant (let's say $\displaystyle a$). Then it is similar to differentiating $\displaystyle \frac{\,d}{\,du}\left[a^u\right]$.
So it follows that $\displaystyle \frac{\partial}{\partial t}\left[x^{f\!\left(t\right)}\right]=x^{f\!\left(t\right)}\ln x\cdot\frac{\,df}{\,dt}$.
Can you take it from here?