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To find the derivative of the function \( F(x) = (x^4 + 9x^2 - 9)^7 \), we will use the **chain rule**. Here's the step-by-step process:

### Step 1: Apply the Chain Rule
The chain rule states that for a function of the form \( F(x) = (g(x))^n \), the derivative is:
\[
F'(x) = n \cdot (g(x))^{n-1} \cdot g'(x)
\]
In this case, \( g(x) = x^4 + 9x^2 - 9 \) and \( n = 7 \).

### Step 2: Differentiate the Outer Function
We differentiate the outer function first:
\[
F'(x) = 7 \cdot (x^4 + 9x^2 - 9)^6
\]

### Step 3: Differentiate the Inner Function \( g(x) = x^4 + 9x^2 - 9 \)
Now, we differentiate \( g(x) \):
\[
g'(x) = 4x^3 + 18x
\]

### Step 4: Multiply the Derivatives
Using the chain rule:
\[
F'(x) = 7 \cdot (x^4 + 9x^2 - 9)^6 \cdot (4x^3 + 18x)
\]

Thus, the derivative is:
\[
F'(x) = 7 \cdot (x^4 + 9x^2 - 9)^6 \cdot (4x^3 + 18x)
\]

Would you like more details on any step, or have any questions?

### Related Questions:
1. What is the chain rule and when is it used?
2. How do you differentiate \(x^4\)?
3. What is the derivative of \(9x^2\)?
4. Can the chain rule be applied multiple times for nested functions?
5. How do you simplify higher-order derivatives of composite functions?

### Tip:
Always start by identifying the inner and outer functions when applying the chain rule.

Find the derivative of the function F(x) = (x^4 + 9x^2 - 9)^7.

Learn how to find the derivative of the function F(x) = (x^4 + 9x^2 - 9)^7 using the chain rule. This detailed step-by-step solution walks through the outer and inner function derivatives, suitable for students in AP Calculus or early college-level calculus.

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