first of all I hope you understand that the first is your propagator:
K=<q' t' | q t> = <q'| e^{-\frac{i}{\hbar} H (t'-t)} |q>=<q'| e^{-\frac{i}{\hbar} (\frac{p^{2}}{2m} + V(q)) (t'-t)} |q>
K(q',q;δt) \approx <q'| [1- \frac{i}{{\hbar}} δt (\frac{p^{2}}{2m}+V(q) ) +...] |q>+... \approx δ(q'-q) - \frac{i}{2m{\hbar}} <q'|p^{2}|q> - \frac{i}{{\hbar}} <q'|V(q)|q>+...
K(q',q;δt) \approx <q|(1- \frac{i}{{\hbar}} \frac{p^{2}}{2m}+...) ( 1- \frac{i}{{\hbar}} V(\frac{q'+q}{2})+...)|q'>
neglecting the terms of order δt2 and more, the propagator is:
K(q',q;δt) \approx {K}_{0} (q',q;δt) e^{-\frac{i}{{\hbar}} V(\frac{q'+q}{2})}
Here I will use a property of the propagator which if you like I can prove you too...
K(q',q;t',t) = {\prod}_{j=1}^{N-1} \int ({dq}_{j} (\frac{m}{2πi{\hbar}δt})^{1/2} e^{\frac{im}{2{\hbar}δt} ({q}_{j} -{q}_{j-1})^{2} -\frac{i}{{\hbar}} δt V(\frac{({q}_{j}+{q}_{j-1})}{2})}
where now
δt= {t}_{j}-{t}_{j-1}/N = (t'-t)/N\rightarrow0 for N\rightarrow∞
the constant infinitesimal time interval step.
for it going to zero, you can insert a function q(t) which takes the values q(tj)=qj in the interval [t',t]. The boundary values of this function is q(t')=q' and q(t)=q. Since each of the integrating coordinates qj takes values from [-∞,∞], the function q(t) even for N→∞ is not a priori continuous function. However because of the Gauss form exp[...(qj-qj-1)2/δt ...] of the integrating function, while δt→0, only the neighbouring points qj, qj-1 contribute importantly at the integration. So, at the limit δt→0, the q(t) is practically a continuous function. Assuming that at this limit we can define derivative we have:
\dot{q}= lim_{δt\rightarrow0} (\frac{q(t+δt)-q(t)}{δt})= lim_{N\rightarrow∞} (\frac{{q}_{j}-{q}_{j-1}}{{t}_{j}-{t}_{j-1}})
for each j. This derivative will replace in the continuous limit the differences that appear on the exponential of the integrating form of the propagator. Also, the sum can change into integral by the rule:
δt {\sum}_{j=1}^{j=N-1} \rightarrow {\int}_{t}^{t'} dτ
at the continuum limit the total exponent takes the form of the Classical Action S for the interval [t',t]:
{\sum}_{j=1}^{j=N-1} [\frac{im}{2{\hbar}δt} ({q}_{j} -{q}_{j-1})^{2} -\frac{i}{{\hbar}} δt V(\frac{({q}_{j}+{q}_{j-1})}{2}) ]→ \frac{i}{{\hbar}} {\int}_{t}^{t'} dτ [\frac{m}{2}\dot{q}(τ)^{2} -V(q(τ)) ]
Where you see it's the Action...
Putting then the integral on the expression of the propagator you get:
K(q',q;t',t) = {\int}_{q(t)=q}^{q(t')=q'} [dq] e^{\frac{i}{{\hbar}} {S}_{c}[q;t',t]}
where [dq] is:
[dq]= {\prod}_{j=1}^{N-1} {dq}_{j} (\frac{m}{2πi{\hbar}δt})^{1/2}
