$$K = \Bigg|\frac{dT}{ds}\Bigg| = \Bigg|\frac{\frac{dT}{dt}}{\frac{ds}{dt}}\Bigg|$$
where $T$ is the unit tangent vector to the curve and $s$ is arc lenght.
I don't know why that relationship is True in infinitesimals, in algebra I know the ratio $\frac{1}{dt}$ both in the numerator and the denominator is one but I am not sure how that works in infinitesimal and with an absolute value
You have $s=s(t)$ and $T=T(s)=T(s(t))$.
Applying the chain rule you get $\frac{dT}{dt}=\frac{dT}{ds}\frac{ds}{dt}$ and then divide by $\frac{ds}{dt}$ to obtain your equality.