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To find \( f'(x) \), we will use the definition of the derivative:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
\]

Given that \( f(x+h) - f(x) = -5hx^2 - 8hx + 6h^2x - 3h^2 + 7h^3 \), we can plug this into the formula:

\[
f'(x) = \lim_{h \to 0} \frac{-5hx^2 - 8hx + 6h^2x - 3h^2 + 7h^3}{h}.
\]

Now, simplify the expression by dividing each term by \( h \):

\[
f'(x) = \lim_{h \to 0} (-5x^2 - 8x + 6hx - 3h + 7h^2).
\]

As \( h \to 0 \), the terms involving \( h \), \( h^2 \), and \( h^3 \) vanish, leaving us with:

\[
f'(x) = -5x^2 - 8x.
\]

So, the derivative is:

\[
f'(x) = -5x^2 - 8x.
\]

Would you like further details or have any questions? Here are five related questions to explore:

1. What is the general process of finding a derivative using the limit definition?
2. How do higher powers of \( h \) disappear as \( h \to 0 \)?
3. Can you find the second derivative \( f''(x) \) from the expression we derived?
4. How does this method compare to using basic differentiation rules (like the power rule)?
5. How does the behavior of \( f'(x) \) change as \( x \) increases or decreases?

**Tip:** When simplifying a derivative expression, always carefully consider which terms will vanish as \( h \to 0 \) to avoid unnecessary complexity.

Suppose that f(x+h) - f(x) = -5hx^2 - 8hx + 6h^2x - 3h^2 + 7h^3. Find f'(x).

Learn how to find the derivative of a function using the limit definition. Given f(x+h) - f(x) = -5hx^2 - 8hx + 6h^2x - 3h^2 + 7h^3, we simplify the expression to find f'(x) by applying limits and analyzing the behavior of higher powers of h.

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