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The problem you provided is asking to find \( f'(x) \), the derivative of the function \( f(x) \), given that:

\[
f(x + h) - f(x) = -8h x^2 + 1h x - 8h^2 x - 3h^2 + 3h^3.
\]

This expression can be interpreted as the difference quotient, \( \frac{f(x+h) - f(x)}{h} \), from which we can derive \( f'(x) \).

### Step-by-step Solution:

1. **Start with the definition of the derivative:**
   \[
   f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
   \]

2. **Divide the given expression by \( h \):**
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{-8h x^2 + 1h x - 8h^2 x - 3h^2 + 3h^3}{h}.
   \]

3. **Simplify the expression:**
   - The first term simplifies to \( -8x^2 \).
   - The second term simplifies to \( 1x = x \).
   - The third term becomes \( -8h x \).
   - The fourth term simplifies to \( -3h \).
   - The fifth term becomes \( 3h^2 \).

   Therefore, the expression simplifies to:
   \[
   \frac{f(x+h) - f(x)}{h} = -8x^2 + x - 8h x - 3h + 3h^2.
   \]

4. **Take the limit as \( h \to 0 \):**
   As \( h \to 0 \), any terms involving \( h \) will vanish. Specifically:
   - The term \( -8h x \) goes to 0.
   - The term \( -3h \) goes to 0.
   - The term \( 3h^2 \) goes to 0.

   Therefore, the limit is:
   \[
   f'(x) = -8x^2 + x.
   \]

### Final Answer:
\[
f'(x) = -8x^2 + x.
\]

Would you like further details on any step or have any questions? Here are five related questions to deepen your understanding:

1. How does the difference quotient help in finding derivatives?
2. What happens to higher-order terms of \( h \) when taking the limit as \( h \to 0 \)?
3. Can you apply the same process to find \( f'(x) \) for a different function \( f(x) \)?
4. What would the second derivative \( f''(x) \) be for this function?
5. How can you use this method to find derivatives for more complex functions?

**Tip:** When dealing with limits in derivative calculations, remember that any term involving \( h \) will vanish as \( h \to 0 \).

Suppose that f(x + h) - f(x) = -8hx^2 + 1hx - 8h^2x - 3h^2 + 3h^3. Find f'(x).

Learn how to find the derivative of a function using the limit definition, based on the equation f(x + h) - f(x) = -8hx^2 + 1hx - 8h^2x - 3h^2 + 3h^3. Step-by-step solution using calculus concepts, limits, and derivatives. Suitable for high school or early college students.

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