The population of bacteria (in millions) in a certain culture $\mathrm{x}$ hours after an experimental nutrient is introduced into the culture is $P(x)=\frac{20 x}{5+x^{2}}$. Use the differential to approximate the changes in population for the following changes in $x$.
a. 1 to 1.5
b. 4 to 4.25
a. Use the differential to approximate the change in population for $x=1$ to 1.5 .
Between 1 and 1.5 hours, the population of bacteria changes by million. (Round to three decimal places as needed.)

Step 1 ：The problem is asking for the change in population of bacteria from $x=1$ to $x=1.5$. This can be approximated using the differential of the function $P(x)$, which represents the rate of change of the population with respect to time. The differential of a function is given by its derivative. So, first we need to find the derivative of $P(x)$, then we can use this to approximate the change in population.

Step 2 ：Let's find the derivative of $P(x)$. Given $P(x)=\frac{20x}{5+x^{2}}$, the derivative $P'(x)$ is calculated as $P'(x) = -\frac{40x^{2}}{(x^{2} + 5)^{2}} + \frac{20}{x^{2} + 5}$.

Step 3 ：Next, we calculate the change in $x$, which is $\Delta x = 1.5 - 1 = 0.5$.

Step 4 ：Then, we use the derivative to approximate the change in population, $\Delta P = P'(1) \cdot \Delta x$.

Step 5 ：By substituting the values into the equation, we get $\Delta P = 1.111$ million.

Step 6 ：Final Answer: The approximate change in population of bacteria from $x=1$ to $x=1.5$ is \(\boxed{1.111}\) million.