I'm trying to prove that if $\vec{x}:I\rightarrow\mathbb{R}^2$ is a curve parametrized by arc length and $\theta(t)$ is the angle between the tangent line to $\vec{x}$ at point $t$ and the $x$ axis, then $\kappa=\theta'$, where $\kappa$ denotes the curvature. 

I know that for a curve in $\mathbb{R}^2$, the curvature is given by 

$\kappa=\frac{1}{|\vec{x}'|^3}det(\vec{x}'\vec{x}'')$. 

I also know that the tangent line to $\vec{x}$ at point $\alpha\in\mathbb{R}$ is given by $y(t)=\vec{x}(\alpha)+t\vec{x}'$. 

I can picture the problem, but can't write the solutions. Any ideas?