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To find the derivative \( f'(x) \) of the function \( f(x) = 7x^3 \ln x \), we will apply the **product rule** because the function is a product of two functions: \( 7x^3 \) and \( \ln x \).

### Steps:
1. **Product Rule**: If \( f(x) = u(x) v(x) \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \).
   - Here, \( u(x) = 7x^3 \) and \( v(x) = \ln x \).

2. **Differentiate each function**:
   - \( u'(x) = \frac{d}{dx} [7x^3] = 21x^2 \).
   - \( v'(x) = \frac{d}{dx} [\ln x] = \frac{1}{x} \).

3. **Apply the product rule**:
   \[
   f'(x) = u'(x)v(x) + u(x)v'(x)
   \]
   \[
   f'(x) = (21x^2)(\ln x) + (7x^3)\left(\frac{1}{x}\right)
   \]

4. **Simplify**:
   \[
   f'(x) = 21x^2 \ln x + 7x^2
   \]

Thus, the derivative is:
\[
f'(x) = 21x^2 \ln x + 7x^2
\]

Would you like further details or have any questions?

Here are five related questions to expand on this concept:
1. How would the derivative change if the function were \( f(x) = 7x^3 \ln(x^2) \)?
2. What happens if you replace \( \ln x \) with \( \log x \) in the original function?
3. Can you apply the product rule to a function like \( f(x) = e^x \ln x \)?
4. How would you find the second derivative, \( f''(x) \), of the same function?
5. What is the physical interpretation of this derivative in a real-world context?

**Tip**: Always simplify each part of a product or quotient before applying differentiation rules to avoid errors.

For the function f(x) = 7x^3 ln x, find f'(x).

Learn how to differentiate the function f(x) = 7x^3 ln x using the product rule. This step-by-step solution covers the differentiation of logarithmic functions and power functions, suitable for students in Calculus I (Grades 11-12).

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