To find the derivative of the function $(3x + 1)^3$, we apply the chain rule. The chain rule is used when differentiating a composite function.

Given:

$y = (3x + 1)^3$

Step 1: Identify the outer and inner functions

• The outer function is $u^3$, where $u = 3x + 1$.
• The inner function is $u = 3x + 1$.

Step 2: Differentiate the outer function with respect to $u$

The derivative of $u^3$ with respect to $u$ is: $\frac{d}{du}(u^3) = 3u^2$

Step 3: Differentiate the inner function with respect to $x$

The derivative of $u = 3x + 1$ with respect to $x$ is: $\frac{d}{dx}(3x + 1) = 3$

Step 4: Apply the chain rule

The chain rule states: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

So, $\frac{d}{dx}[(3x + 1)^3] = 3(3x + 1)^2 \times 3$

Step 5: Simplify

$\frac{d}{dx}[(3x + 1)^3] = 9(3x + 1)^2$

Thus, the derivative of $(3x + 1)^3$ with respect to $x$ is:

$\boxed{9(3x + 1)^2}$

Would you like further details or have any additional questions?

Relative Questions:

1. What is the chain rule, and when is it used?
2. How does the chain rule apply to functions of functions?
3. What is the importance of identifying inner and outer functions in the chain rule?
4. Can you differentiate more complex functions using the chain rule?
5. What happens if the inner function is more complicated, such as a product or quotient?

Tip: Always clearly identify the inner and outer functions when applying the chain rule to avoid mistakes.