Find 

f prime left parenthesis x right parenthesisf′(x).

f left parenthesis x right parenthesis equals left parenthesis 2 x Superscript 6 Baseline plus 9 right parenthesis Superscript 5f(x)=2x6+95

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Part 1

f prime left parenthesis x right parenthesisf′(x)equals=enter your response here

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.

Learn how to differentiate the function f(x) = (2x^6 + 9)^5 using the chain rule of differentiation with a detailed step-by-step solution. This problem covers differentiation techniques for composite functions and the chain rule formula. Suitable for students in Grades 11-12.

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