We are tasked to find the derivative of $u = \left( \ln(2x) \right)^2$ with respect to $x$.

Step-by-step Solution:

Step 1: Use the chain rule

The function is a composition of functions. First, let $u = \left( f(x) \right)^2$, where $f(x) = \ln(2x)$. Using the chain rule: $\frac{du}{dx} = 2 f(x) \cdot \frac{df(x)}{dx}.$

Step 2: Differentiate $f(x) = \ln(2x)$

We know: $\frac{d}{dx} \ln(2x) = \frac{1}{2x} \cdot 2 = \frac{1}{x}.$

Thus: $\frac{df(x)}{dx} = \frac{1}{x}.$

Step 3: Substitute $f(x)$ and $\frac{df(x)}{dx}$ into the chain rule

Now substitute back: $\frac{du}{dx} = 2 \ln(2x) \cdot \frac{1}{x}.$

Final Answer:

$\frac{du}{dx} = \frac{2 \ln(2x)}{x}.$

Let me know if you need a detailed explanation for any step!

Follow-up Questions:

1. What is the interpretation of $\ln(2x)$ in this context?
2. How does the chain rule apply to nested functions like this one?
3. How would the derivative change if $u = (\ln(2x))^3$?
4. What happens to the derivative as $x \to \infty$?
5. Can you evaluate $\frac{du}{dx}$ at a specific value of $x$, such as $x = 1$?

Tip:

When dealing with logarithmic derivatives, always remember to simplify the arguments of the logarithm if possible to avoid confusion during differentiation!