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To find the derivative of the given function \( f(x) = \frac{3x - 4}{x^3 - 5x + 10} \), we can apply the **quotient rule** for differentiation. 

The quotient rule states that for a function \( f(x) = \frac{g(x)}{h(x)} \), the derivative \( f'(x) \) is given by:

\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
\]

Here, \( g(x) = 3x - 4 \) and \( h(x) = x^3 - 5x + 10 \).

### Step 1: Differentiate \( g(x) \) and \( h(x) \)
1. \( g(x) = 3x - 4 \)
   \[
   g'(x) = 3
   \]

2. \( h(x) = x^3 - 5x + 10 \)
   \[
   h'(x) = 3x^2 - 5
   \]

### Step 2: Apply the quotient rule
Using the quotient rule formula:

\[
f'(x) = \frac{(3)(x^3 - 5x + 10) - (3x - 4)(3x^2 - 5)}{(x^3 - 5x + 10)^2}
\]

### Step 3: Simplify the expression
Now expand both the numerator and denominator.

- First, expand \( 3(x^3 - 5x + 10) \):
  \[
  3(x^3 - 5x + 10) = 3x^3 - 15x + 30
  \]

- Next, expand \( (3x - 4)(3x^2 - 5) \):
  \[
  (3x - 4)(3x^2 - 5) = 3x(3x^2 - 5) - 4(3x^2 - 5)
  \]
  \[
  = 9x^3 - 15x - 12x^2 + 20
  \]

Now, combine these terms in the numerator:

\[
f'(x) = \frac{(3x^3 - 15x + 30) - (9x^3 - 12x^2 - 15x + 20)}{(x^3 - 5x + 10)^2}
\]
\[
= \frac{3x^3 - 15x + 30 - 9x^3 + 12x^2 + 15x - 20}{(x^3 - 5x + 10)^2}
\]
\[
= \frac{-6x^3 + 12x^2 + 10}{(x^3 - 5x + 10)^2}
\]

### Final answer:
\[
f'(x) = \frac{-6x^3 + 12x^2 + 10}{(x^3 - 5x + 10)^2}
\]

Would you like me to explain any steps in more detail? 

Here are 5 related questions to think about:
1. How does the quotient rule differ from the product rule?
2. Can you apply the chain rule in this example? If so, where?
3. What would happen if the denominator in the original function had a higher degree?
4. How can we verify that this derivative is correct?
5. What are other real-world examples where the quotient rule is applied?

**Tip**: Always simplify the numerator carefully when applying the quotient rule to avoid mistakes in the expansion process!

Find f'(x) for the function f(x) = (3x - 4) / (x^3 - 5x + 10).

This problem explains how to find the derivative of the function f(x) = (3x - 4) / (x^3 - 5x + 10) using the quotient rule. Step-by-step solution with differentiation of the numerator and denominator. Suitable for advanced high school or college-level calculus students.

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