At time $t = 0$, a projectile of mass $m$ is launched from the origin at an angle $α$ to the horizontal with speed $U$. Let the position vector of the projectile be $\mathbf{r} = x\mathbf{i}+z\mathbf{k}$.
Determine an expression for the speed $v$ of the projectile at time $t$.
I have worked out already that $x = Ut\cos\alpha$ and $z = -\frac{1}{2}gt^2+Ut\sin \alpha$.
So, this means $\mathbf{r}=Ut\cos\alpha\mathbf{i}+(-\frac{1}{2}gt^2+Ut\sin \alpha)\mathbf{k}$.
Am I correct in differentiating $\mathbf{r}$ with respect to time $t$ and then calculating the magnitude using Pythagoras to calculate speed $u$? I am not sure whether I differentiated correctly:
$\mathbf{\dot{r}}=U\cos\alpha\mathbf{i}+(-gt+U\sin \alpha)\mathbf{k}$, So, $u = \sqrt{U^2 \cos{\alpha}^2+(-gt+U\sin \alpha)^2}$?
David K answered your question, but for the sake of completeness, I am posting an answer.
As you can see from this Wikipedia article, your reasoning and computations are correct.