Discrete State Space Model - Why Are We Calculating $ x \left[ k + 1 \right] $ Instead of $ \dot{\boldsymbol{x}} \left( t \right) $? A continuous state space model is defined as follows.
$$
\dot{\boldsymbol{x}}(t) = A \boldsymbol{x}(t)+ B \boldsymbol{u}(t) \\
\boldsymbol{y}(t)= C \boldsymbol{x}(t)+ D \boldsymbol{u}(t)
$$

If we want to calculate the state $\boldsymbol{x}$ for a specific time $t$, we need to use the state transition matrix $\Phi$. However, when looking at a discrete state space model, things look different.
$$
\boldsymbol{x}[k+1] = {A}_{d} \boldsymbol{x}[k] + {B}_{d} \boldsymbol{u}[k] \\
\boldsymbol{y}[k] = C \boldsymbol{x}[k] + D \boldsymbol{u}[k]
$$

With the discrete state space model, there seem to be no 1st order differential equations incorporated anymore. Instead, we can directly calculate the next sample, without the need for a state transition matrix. How come this is the case?
I will ask you something that will give you intuition.
How would you calculate the Gradient of an image?
Image is a discretization of reality, so how would you estimate the gradient of the "Reality" if you're given only the image?  

In the case above we use Finite Differences to approximate the continuous derivative.
So what actually is approximating $ \dot{x} \left( t \right) $ is:

$$ \dot{x} \left( k \Delta t\right) \approx \frac{ x \left[ k + 1 \right] - x \left[ k \right]}{ \Delta t} $$

I do a little abuse of the $ k $ parameter yet I think you can understand what is going on there.