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We are tasked with calculating \( F'(0) \), where

\[
F(x) = \frac{6x^7 - 10x^9 + x^5 - 6x}{8x^8 - 4x^9 + 10x^3 - 10}.
\]

The hint suggests expressing \( F(x) \) as \( F(x) = \frac{f(x)}{g(x)} \), where:

\[
f(x) = 6x^7 - 10x^9 + x^5 - 6x
\]
and
\[
g(x) = 8x^8 - 4x^9 + 10x^3 - 10.
\]

To calculate \( F'(x) \) using the quotient rule, we need the derivatives of \( f(x) \) and \( g(x) \):

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

We are asked to compute \( F'(0) \), so let’s first evaluate \( f(0) \), \( f'(0) \), \( g(0) \), and \( g'(0) \).

### Step 1: Evaluate \( f(0) \) and \( g(0) \)

\[
f(0) = 6(0)^7 - 10(0)^9 + (0)^5 - 6(0) = 0,
\]
\[
g(0) = 8(0)^8 - 4(0)^9 + 10(0)^3 - 10 = -10.
\]

### Step 2: Find \( f'(x) \) and \( g'(x) \)

\[
f'(x) = 42x^6 - 90x^8 + 5x^4 - 6,
\]
\[
f'(0) = 42(0)^6 - 90(0)^8 + 5(0)^4 - 6 = -6,
\]
\[
g'(x) = 64x^7 - 36x^8 + 30x^2,
\]
\[
g'(0) = 64(0)^7 - 36(0)^8 + 30(0)^2 = 0.
\]

### Step 3: Apply the quotient rule

At \( x = 0 \),
\[
F'(0) = \frac{f'(0)g(0) - f(0)g'(0)}{[g(0)]^2}.
\]

Substituting the values:
\[
F'(0) = \frac{(-6)(-10) - (0)(0)}{(-10)^2} = \frac{60}{100} = 0.6.
\]

Thus, the value of \( F'(0) \) is \( 0.6 \).

---

Would you like further details or clarification on any step? Here are some related questions you might find interesting:

1. How is the quotient rule derived in calculus?
2. Can the quotient rule be applied when both \( f(x) \) and \( g(x) \) are non-polynomial functions?
3. What happens if \( g(0) = 0 \) in a problem like this?
4. How can we apply L'Hôpital's Rule in indeterminate forms involving \( \frac{0}{0} \)?
5. How does the behavior of higher-order terms influence the result at specific points?

**Tip:** When dealing with rational functions, always check whether \( g(x) \) is zero at the points you're interested in, as it may affect the result.

Calculate F'(0), where F(x) = (6x^7 - 10x^9 + x^5 - 6x) / (8x^8 - 4x^9 + 10x^3 - 10). Express F'(0) directly in terms of f(0), f'(0), g(0), g'(0).

This problem explores the application of the quotient rule to calculate F'(0), where F(x) = (6x^7 - 10x^9 + x^5 - 6x) / (8x^8 - 4x^9 + 10x^3 - 10). Detailed steps include finding f(0), g(0), f'(0), and g'(0), followed by applying the quotient rule.

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