To find the derivative of the function $f(x) = \frac{1}{3x+2}$ using the chain rule, follow these steps:

Step 1: Identify the Outer and Inner Functions

We can rewrite the function $f(x)$ as:

$f(x) = (3x + 2)^{-1}$

Here, the function is composed of two functions:

• The outer function is $g(u) = u^{-1}$.
• The inner function is $u(x) = 3x + 2$.

Step 2: Differentiate the Outer Function with Respect to the Inner Function

Differentiate $g(u) = u^{-1}$ with respect to $u$:

$g'(u) = -u^{-2} = -\frac{1}{u^2}$

Step 3: Differentiate the Inner Function with Respect to $x$

Differentiate $u(x) = 3x + 2$ with respect to $x$:

$u'(x) = 3$

Step 4: Apply the Chain Rule

The chain rule states:

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

Substitute $u(x) = 3x + 2$ and the derivatives we calculated:

$f'(x) = -\frac{1}{(3x + 2)^2} \cdot 3$

Step 5: Simplify the Expression

Finally, simplify the expression:

$f'(x) = \frac{-3}{(3x + 2)^2}$

So, the derivative of $f(x) = \frac{1}{3x+2}$ using the chain rule is:

$f'(x) = \frac{-3}{(3x + 2)^2}$

Would you like further explanation or have any questions?

Here are 8 related questions you might consider:

1. How does the chain rule simplify the process of finding derivatives of composite functions?
2. Can you find the derivative of $f(x) = \frac{1}{5x + 7}$ using the chain rule?
3. How can the chain rule be applied to a function like $f(x) = \frac{1}{(2x + 3)^3}$?
4. What is the derivative of $f(x) = \frac{1}{3x+2}$ using the quotient rule?
5. How would the derivative change if $f(x) = \frac{1}{\sqrt{3x+2}}$?
6. What role does the chain rule play in finding the derivative of implicit functions?
7. How can the chain rule be extended to functions of multiple variables?
8. Can you find the second derivative of $f(x) = \frac{1}{3x + 2}$ using the chain rule?

Tip: When using the chain rule, clearly identify the inner and outer functions to ensure accuracy in differentiation.