To evaluate \( f'(3) \) for the function \( f(x) = \frac{1}{7}x^3 + \frac{5}{9}x^5 \), we need to follow these steps:

1. **Differentiate the function \( f(x) \).**
2. **Evaluate the derivative at \( x = 3 \).**

### Step 1: Differentiate the function

Given \( f(x) = \frac{1}{7}x^3 + \frac{5}{9}x^5 \), we will differentiate term by term using the power rule, which states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

\[ f(x) = \frac{1}{7}x^3 + \frac{5}{9}x^5 \]

Differentiate each term:
\[ \left(\frac{1}{7}x^3\right)' = \frac{1}{7} \cdot 3x^2 = \frac{3}{7}x^2 \]
\[ \left(\frac{5}{9}x^5\right)' = \frac{5}{9} \cdot 5x^4 = \frac{25}{9}x^4 \]

So the derivative \( f'(x) \) is:
\[ f'(x) = \frac{3}{7}x^2 + \frac{25}{9}x^4 \]

### Step 2: Evaluate the derivative at \( x = 3 \)

\[ f'(3) = \frac{3}{7}(3^2) + \frac{25}{9}(3^4) \]

Calculate the powers of 3:
\[ 3^2 = 9 \]
\[ 3^4 = 81 \]

Substitute these values in:
\[ f'(3) = \frac{3}{7}(9) + \frac{25}{9}(81) \]
\[ = \frac{27}{7} + \frac{2025}{9} \]

Convert the second term to have a common denominator with the first term:
\[ \frac{2025}{9} = 225 \]
\[ f'(3) = \frac{27}{7} + 225 \]

Convert 225 to a fraction with a denominator of 7:
\[ 225 = \frac{1575}{7} \]

Add the fractions:
\[ f'(3) = \frac{27}{7} + \frac{1575}{7} \]
\[ = \frac{1602}{7} \]

Thus, \( f'(x) \) evaluated at \( x = 3 \) is \( \frac{1602}{7} \) or approximately 228.857.

Would you like any further details or have any questions?

Here are 8 related questions that you might find interesting:

1. How does the power rule for differentiation work?
2. Can you provide an example of differentiating a polynomial with more terms?
3. What is the process for evaluating derivatives at specific points?
4. How would the chain rule apply if \( f(x) \) were a composite function?
5. Can you explain the differences between the power rule and the product rule?
6. How would you find the second derivative of \( f(x) \)?
7. How can you verify the result using a numerical approximation method?
8. What are the practical applications of finding derivatives at specific points?

**Tip:** When working with fractions, always look for opportunities to simplify or find common denominators early to make calculations easier.

Learn how to evaluate the derivative f'(3) for the function f(x) = (1/7)x^3 + (5/9)x^5. Detailed step-by-step solution using differentiation and power rule. Suitable for advanced high school students in Grades 11-12.

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