The number of bacteria $P$ is modeled by the equation:

$P(t) = 500 \left( 4 + \frac{5t}{40 + t^2} \right)$

We are tasked with finding $P'(t)$, the derivative of $P(t)$ with respect to $t$.

Step 1: Differentiate the constant multiplier

The constant factor $500$ can be factored out of the differentiation, so we first focus on differentiating the inner function:

$P(t) = 500 \left( 4 + \frac{5t}{40 + t^2} \right)$

Let $g(t) = 4 + \frac{5t}{40 + t^2}$. Therefore:

$P'(t) = 500 \cdot g'(t)$

Step 2: Differentiate the inner function $g(t) = 4 + \frac{5t}{40 + t^2}$

First, observe that the constant term $4$ has a derivative of zero:

$\frac{d}{dt} [4] = 0$

Now, we differentiate the second term $\frac{5t}{40 + t^2}$ using the quotient rule. The quotient rule states:

$\frac{d}{dt} \left( \frac{u(t)}{v(t)} \right) = \frac{u'(t) v(t) - u(t) v'(t)}{[v(t)]^2}$

For the term $\frac{5t}{40 + t^2}$:

• $u(t) = 5t$, so $u'(t) = 5$
• $v(t) = 40 + t^2$, so $v'(t) = 2t$

Now apply the quotient rule:

$\frac{d}{dt} \left( \frac{5t}{40 + t^2} \right) = \frac{5(40 + t^2) - 5t(2t)}{(40 + t^2)^2}$

Simplifying the numerator:

$= \frac{5(40 + t^2) - 10t^2}{(40 + t^2)^2} = \frac{200 + 5t^2 - 10t^2}{(40 + t^2)^2} = \frac{200 - 5t^2}{(40 + t^2)^2}$

Thus, the derivative of $g(t)$ is:

$g'(t) = \frac{200 - 5t^2}{(40 + t^2)^2}$

Step 3: Multiply by 500

Finally, multiply $g'(t)$ by 500 to get $P'(t)$:

$P'(t) = 500 \cdot \frac{200 - 5t^2}{(40 + t^2)^2}$

Simplifying:

$P'(t) = \frac{100000 - 2500t^2}{(40 + t^2)^2}$

Final Answer:

$P'(t) = \frac{100000 - 2500t^2}{(40 + t^2)^2}$

Let me know if you'd like further details, or if you have any questions!

---

Related Questions:

1. How can the derivative help in analyzing population growth over time?
2. What is the meaning of the quotient rule in this context?
3. Can this population growth model be used for long-term predictions? Why or why not?
4. How would we find the time at which the bacteria population is increasing most rapidly?
5. What other applications could this type of growth model have?

Tip: When dealing with fractions, the quotient rule is essential for finding derivatives correctly!