We are tasked with finding the derivative of the function:

$q = \sqrt{19r - r^7}$

Step 1: Rewrite the function

Express the square root in terms of exponents to make differentiation easier:

$q = (19r - r^7)^{1/2}$

Step 2: Apply the chain rule

To differentiate this, we'll apply the chain rule. The chain rule states:

$\frac{d}{dr} \left[ f(g(r)) \right] = f'(g(r)) \cdot g'(r)$

Here, $f(u) = u^{1/2}$ and $g(r) = 19r - r^7$.

Step 3: Differentiate the outer function

Differentiate $f(u) = u^{1/2}$:

$\frac{d}{du} \left[ u^{1/2} \right] = \frac{1}{2} u^{-1/2}$

Thus, applying this to $(19r - r^7)^{1/2}$:

$\frac{d}{dr} \left[ (19r - r^7)^{1/2} \right] = \frac{1}{2} (19r - r^7)^{-1/2} \cdot \frac{d}{dr} (19r - r^7)$

Step 4: Differentiate the inner function

Now, differentiate $g(r) = 19r - r^7$:

$\frac{d}{dr} (19r - r^7) = 19 - 7r^6$

Step 5: Combine results

Now combine the outer and inner derivatives:

$\frac{dq}{dr} = \frac{1}{2} (19r - r^7)^{-1/2} \cdot (19 - 7r^6)$

This is the derivative of the given function.

Would you like further explanation or additional details on any step?

Related questions:

1. What is the chain rule, and how does it work in differentiation?
2. How do we handle square roots in differentiation problems?
3. Can you explain the process of differentiating powers of variables?
4. What happens if we apply the product or quotient rule to complex derivatives?
5. How do we evaluate the derivative at specific values of $r$?

Tip: When working with square roots in calculus, it's often helpful to convert them into exponent form to simplify the differentiation process.