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We are given the function \( q(x) = 7e^x \ln(x) \), and asked to find \( q'(6) \) to the nearest tenth.

### Step 1: Differentiate \( q(x) \)
The function involves a product of two functions: \( 7e^x \) and \( \ln(x) \). We can apply the **product rule** for differentiation, which states:
\[
(fg)' = f'g + fg'
\]
Here, we let:
- \( f(x) = 7e^x \)
- \( g(x) = \ln(x) \)

First, differentiate each component:
1. \( f'(x) = 7e^x \) because the derivative of \( e^x \) is \( e^x \).
2. \( g'(x) = \frac{1}{x} \) because the derivative of \( \ln(x) \) is \( \frac{1}{x} \).

Now, apply the product rule:
\[
q'(x) = f'(x)g(x) + f(x)g'(x)
\]
Substituting in the functions:
\[
q'(x) = 7e^x \ln(x) + 7e^x \cdot \frac{1}{x}
\]
Simplifying:
\[
q'(x) = 7e^x \ln(x) + \frac{7e^x}{x}
\]

### Step 2: Evaluate \( q'(6) \)
Substitute \( x = 6 \) into the derivative expression:
\[
q'(6) = 7e^6 \ln(6) + \frac{7e^6}{6}
\]

We can now compute the numerical values:
- \( e^6 \approx 403.43 \)
- \( \ln(6) \approx 1.7918 \)

So:
\[
q'(6) \approx 7 \times 403.43 \times 1.7918 + \frac{7 \times 403.43}{6}
\]
\[
q'(6) \approx 5048.97 + 471.67
\]
\[
q'(6) \approx 5519.64
\]

### Final Answer:
To the nearest tenth, \( q'(6) \approx 5519.6 \).

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

Here are 5 related questions to expand on this:
1. How would the result change if the function were \( q(x) = 7e^{2x} \ln(x) \)?
2. What would the second derivative \( q''(x) \) look like for this function?
3. How does the product rule differ from the quotient rule in calculus?
4. Can you explain the significance of the natural logarithm \( \ln(x) \) in calculus?
5. What happens to \( q'(x) \) as \( x \to 0^+ \)?

**Tip:** When differentiating products, always identify the two functions involved and apply the product rule carefully.

If q(x) = 7e^x ln(x), find q'(6) to the nearest tenth.

Learn how to find the derivative of the function q(x) = 7e^x ln(x) and calculate q'(6) to the nearest tenth. This calculus problem involves applying the product rule and understanding the derivatives of exponential and logarithmic functions. Suitable for students in Grades 11-12.

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