Find the rate of change of total revenue, cost, and profit with respect to time. Assume that $R(x)$ and $C(x)$ are in dollars.
$R(x)=2 x, \quad C(x)=0.01 x^{2}+0.2 x+5$, when $x=30$ and $\frac{d x}{d t}=9$ units per day

Step 1 ：Given the revenue function \(R(x) = 2x\) and the cost function \(C(x) = 0.01x^2 + 0.2x + 5\), we are asked to find the rate of change of total revenue, cost, and profit with respect to time when \(x = 30\) and \(\frac{dx}{dt} = 9\) units per day.

Step 2 ：The rate of change of total revenue, cost, and profit with respect to time can be found by taking the derivative of the revenue and cost functions with respect to x, and then multiplying by the rate of change of x with respect to time.

Step 3 ：The derivative of the revenue function \(R(x)\) with respect to x is \(\frac{dR}{dx} = 2\).

Step 4 ：The derivative of the cost function \(C(x)\) with respect to x is \(\frac{dC}{dx} = 0.02x + 0.2\).

Step 5 ：Multiplying these derivatives by the rate of change of x with respect to time, we get \(\frac{dR}{dt} = 2 * \frac{dx}{dt} = 18\) and \(\frac{dC}{dt} = (0.02x + 0.2) * \frac{dx}{dt} = 0.18x + 1.8\).

Step 6 ：The profit function is the difference between the revenue and cost functions, so its rate of change can be found by subtracting the rate of change of cost from the rate of change of revenue. This gives us \(\frac{dP}{dt} = \frac{dR}{dt} - \frac{dC}{dt} = 16.2 - 0.18x\).

Step 7 ：Substituting \(x = 30\) into these expressions, we find that \(\frac{dR}{dt} = 18\), \(\frac{dC}{dt} = 7.2\), and \(\frac{dP}{dt} = 10.8\).

Step 8 ：Final Answer: The rate of change of total revenue with respect to time when \(x=30\) is \(\boxed{18}\) dollars per day, the rate of change of total cost with respect to time when \(x=30\) is \(\boxed{7.2}\) dollars per day, and the rate of change of total profit with respect to time when \(x=30\) is \(\boxed{10.8}\) dollars per day.