To find the derivative of the function $q = \sqrt{18x - x^7}$, we will apply the chain rule.

Step 1: Express the function in a more convenient form

We can rewrite the function as:

$q = (18x - x^7)^{1/2}$

Step 2: Differentiate using the chain rule

We apply the chain rule, which states that if you have a function of the form $f(g(x))$, then the derivative is:

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

In our case:

• The outer function is $(u)^{1/2}$, where $u = 18x - x^7$.
• The derivative of $(u)^{1/2}$ is $\frac{1}{2} u^{-1/2}$.
• The inner function is $18x - x^7$, and its derivative is $\frac{d}{dx}(18x - x^7) = 18 - 7x^6$.

Step 3: Combine the results

Now, applying the chain rule:

$\frac{dq}{dx} = \frac{1}{2} (18x - x^7)^{-1/2} \cdot (18 - 7x^6)$

Final result:

$\frac{dq}{dx} = \frac{18 - 7x^6}{2\sqrt{18x - x^7}}$

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Let me know if you'd like further clarification or details! Here are some follow-up questions:

1. What is the second derivative of $q = \sqrt{18x - x^7}$?
2. How does the chain rule apply to more complex functions?
3. How do you find the derivative of $\sin(x^2)$?
4. Can you find the derivative of $e^{x^2 + 2x}$?
5. What is the geometric interpretation of a derivative?

Tip: Always double-check whether a function requires a chain rule by looking for compositions of functions—this will help you avoid mistakes in differentiation!