Step 1 :Given the rate of profit function \(P^{\prime}(t)=(7 t+14)\left(t^{2}+4 t+3\right)^{\frac{1}{6}}\)
Step 2 :To find the total profit in the first four years, we need to integrate the rate of profit function from 0 to 4. The integral of a function gives the area under the curve, which in this case represents the total profit over a certain period of time.
Step 3 :Let's denote the time as \(t\), and the rate of profit as \(profit\_rate = (7*t + 14)*(t**2 + 4*t + 3)**0.166666666666667\)
Step 4 :By calculating the integral, we find that the total profit in the first four years is approximately 179.095506461639 thousand dollars.
Step 5 :So, the final answer is: The total profit in the first four years is \(\boxed{179.095506461639}\) thousand dollars.