If $\frac{d}{dt} f(x,t)>\frac{1}{t^{\alpha}}$ for $0< \alpha\le1$, then $f(x,t)$ is not uniformly bounded.

Question

I am trying to a function $f(x,t)$ is not uniformly bounded if $\frac{d}{dt}f(x,t)>\frac{1}{t^{\alpha}}$ for $0< \alpha\le1$.

By definition, I started by assuming that there is a number $C$, which is independent to $t$, such that $|f(x,t)|\le C$ to get contradiction. However, there is no idea to prove it.