calculate the derevatives f(g(x)))' and g(f(x)))' if f(x)=x^9   g(x)=1/(3x+2) express them in exact forms

We are tasked with finding the derivatives of the following composite functions: 

1. \( \left(f(g(x))\right)' \)
2. \( \left(g(f(x))\right)' \)

Given:
- \( f(x) = x^9 \)
- \( g(x) = \frac{1}{3x+2} \)

### 1. Derivative of \( f(g(x)) \)
We use the chain rule to differentiate \( f(g(x)) \). The chain rule states:

\[
\frac{d}{dx} \left(f(g(x))\right) = f'(g(x)) \cdot g'(x)
\]

First, calculate the derivatives of \( f(x) \) and \( g(x) \):

- \( f'(x) = 9x^8 \)
- \( g'(x) = \frac{d}{dx} \left(\frac{1}{3x + 2}\right) = \frac{-3}{(3x + 2)^2} \)

Now, applying the chain rule:

\[
\frac{d}{dx} \left(f(g(x))\right) = f'(g(x)) \cdot g'(x)
\]
\[
f'(g(x)) = 9(g(x))^8 = 9 \left( \frac{1}{3x + 2} \right)^8
\]

So:

\[
\frac{d}{dx} \left(f(g(x))\right) = 9 \left(\frac{1}{3x + 2}\right)^8 \cdot \frac{-3}{(3x + 2)^2}
\]

Simplifying:

\[
\frac{d}{dx} \left(f(g(x))\right) = -27 \cdot \frac{1}{(3x + 2)^{10}}
\]

### 2. Derivative of \( g(f(x)) \)

Similarly, we use the chain rule for \( g(f(x)) \):

\[
\frac{d}{dx} \left(g(f(x))\right) = g'(f(x)) \cdot f'(x)
\]

We already know:
- \( g'(x) = \frac{-3}{(3x + 2)^2} \)
- \( f'(x) = 9x^8 \)

So, applying the chain rule:

\[
\frac{d}{dx} \left(g(f(x))\right) = g'(f(x)) \cdot f'(x)
\]

Now, substitute \( f(x) = x^9 \) into \( g'(x) \):

\[
g'(f(x)) = \frac{-3}{(3f(x) + 2)^2} = \frac{-3}{(3x^9 + 2)^2}
\]

Thus:

\[
\frac{d}{dx} \left(g(f(x))\right) = \frac{-3}{(3x^9 + 2)^2} \cdot 9x^8
\]

Simplifying:

\[
\frac{d}{dx} \left(g(f(x))\right) = \frac{-27x^8}{(3x^9 + 2)^2}
\]

### Final Answers:
1. \( \frac{d}{dx} \left(f(g(x))\right) = \frac{-27}{(3x + 2)^{10}} \)
2. \( \frac{d}{dx} \left(g(f(x))\right) = \frac{-27x^8}{(3x^9 + 2)^2} \)

Would you like any details or have any questions about these derivatives?

Here are five related questions you might find useful:
1. How does the chain rule apply when there are more than two functions composed together?
2. Can you explain the derivative of an implicit function using these principles?
3. What is the derivative of \( (f(x))^n \) for general functions \( f(x) \)?
4. How would these derivatives change if \( g(x) \) had a different form, say \( g(x) = \ln(3x + 2) \)?
5. How do you apply the product rule in conjunction with the chain rule?

**Tip:** When differentiating composite functions, always identify the "inner" and "outer" functions clearly to apply the chain rule effectively.

Calculate the derivatives of f(g(x))' and g(f(x))' if f(x) = x^9 and g(x) = 1/(3x + 2), expressing them in exact forms.

Learn how to calculate the derivatives of composite functions f(g(x)) and g(f(x)) using the chain rule. The problem involves f(x) = x^9 and g(x) = 1/(3x + 2), applying key calculus concepts such as the chain rule and power rule. Detailed step-by-step solutions are provided, suitable for undergraduate calculus students.

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