We are asked to find the derivative of the function:

$q = \sqrt{10r - r^3}$

To find the derivative, we will use the chain rule.

Step 1: Express the square root as an exponent

We rewrite the function in exponent form:

$q = (10r - r^3)^{1/2}$

Step 2: Differentiate using the chain rule

The chain rule states that if $f(r) = (g(r))^{n}$, then $f'(r) = n \cdot (g(r))^{n-1} \cdot g'(r)$.

Here, $g(r) = 10r - r^3$ and $n = \frac{1}{2}$.

So, applying the chain rule:

$\frac{dq}{dr} = \frac{1}{2} \cdot (10r - r^3)^{-1/2} \cdot \frac{d}{dr}(10r - r^3)$

Step 3: Find the derivative of the inner function

Now we differentiate $g(r) = 10r - r^3$:

$\frac{d}{dr}(10r - r^3) = 10 - 3r^2$

Step 4: Combine the results

Putting it all together:

$\frac{dq}{dr} = \frac{1}{2} \cdot (10r - r^3)^{-1/2} \cdot (10 - 3r^2)$

Thus, the derivative of the function is:

$\frac{dq}{dr} = \frac{10 - 3r^2}{2\sqrt{10r - r^3}}$

Would you like more details on this solution? Let me know if you have any questions!

Related Questions:

1. What is the derivative of a function with a cube root instead of a square root?
2. How does the chain rule apply to functions with multiple terms inside the square root?
3. What is the second derivative of $q = \sqrt{10r - r^3}$?
4. Can we use the chain rule for higher powers or other types of functions?
5. What is the physical interpretation of this derivative in terms of rate of change?

Tip: Always check the domain of the original function when differentiating, especially when dealing with roots. For $q = \sqrt{10r - r^3}$, make sure that $10r - r^3 \geq 0$ for valid real values of $r$.