To find $f'(x)$ where $f(x) = \left( 2x^6 + 9 \right)^5$, we need to apply the chain rule of differentiation.

Step 1: Identify the Outer and Inner Functions

In this function, we can identify:

• Outer function: $g(u) = u^5$
• Inner function: $u(x) = 2x^6 + 9$

Step 2: Differentiate the Outer Function

Differentiate the outer function with respect to $u$:

$g'(u) = 5u^4$

Step 3: Differentiate the Inner Function

Differentiate the inner function with respect to $x$:

$u'(x) = \frac{d}{dx} (2x^6 + 9) = 12x^5$

Step 4: Apply the Chain Rule

Using the chain rule, $f'(x) = g'(u) \cdot u'(x)$:

$f'(x) = 5(2x^6 + 9)^4 \cdot 12x^5$

Step 5: Simplify the Expression

Simplifying the expression, we get:

$f'(x) = 60x^5 (2x^6 + 9)^4$

Final Result

Thus, the derivative of the function is:

$f'(x) = 60x^5 (2x^6 + 9)^4$

If you have any further questions or need additional details, feel free to ask!

Here are some related questions to explore this topic further:

1. How does the chain rule help in differentiating composite functions?
2. What are some examples of functions where the chain rule is necessary?
3. How can we verify the correctness of our derivative?
4. How does the power rule for differentiation work?
5. What are some common mistakes to avoid when using the chain rule?
6. How would the derivative change if we modify the inner function?
7. What are the real-world applications of derivatives in physics or engineering?
8. How can we use derivatives to find the maxima or minima of a function?

Tip: When applying the chain rule, always remember to differentiate both the outer and inner functions and multiply their derivatives together.