I have seen the euler equation in discrete time for the baseline neoclassical growth model written as: $$\frac{U'(c_{t+1})}{U'(c_{t})}=\frac{1}{\beta(1+r)}$$
however I have also seen the euler equation for the continous time equivalent written as: $$\frac{U''(c(t))}{U'(c(t))}\dot{c}(t)=r-f'(k)$$
Ignoring the right hand sides of these equations (which vary based on model set up) and exclusively looking at the left hand side I see two very different formulations of the same equation.
My question is: "How does the Euler equation in Continous time represent the same thing conceptually when compared to its discrete time counter part?"