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.