To find the derivative of the given function:

$F(x) = \left( x^4 + 3x^2 - 6 \right)^5$

We will apply the chain rule, which states that the derivative of a composite function $f(g(x))$ is:

$F'(x) = f'(g(x)) \cdot g'(x)$

Step 1: Differentiate the outer function

Let $g(x) = x^4 + 3x^2 - 6$. The outer function is $f(g(x)) = g(x)^5$. The derivative of the outer function with respect to $g(x)$ is:

$\frac{d}{dg(x)} \left( g(x)^5 \right) = 5 \cdot g(x)^4$

Step 2: Differentiate the inner function

Now, differentiate $g(x) = x^4 + 3x^2 - 6$:

$g'(x) = 4x^3 + 6x$

Step 3: Apply the chain rule

Using the chain rule:

$F'(x) = 5 \cdot \left( x^4 + 3x^2 - 6 \right)^4 \cdot \left( 4x^3 + 6x \right)$

Thus, the derivative of the function is:

$F'(x) = 5 \cdot \left( x^4 + 3x^2 - 6 \right)^4 \cdot (4x^3 + 6x)$

Let me know if you'd like further details or clarifications.

Here are 5 related questions to explore:

1. How does the chain rule apply to functions with multiple layers of composition?
2. How would the result change if the exponent outside the parentheses were different?
3. Can you expand the derivative fully by multiplying all terms?
4. How do you apply the chain rule for functions with more complex inner functions?
5. What happens if the inner function has more terms or involves trigonometric functions?

Tip: Always identify the inner and outer functions clearly when using the chain rule to avoid mistakes.