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.