To find the derivative of the expression

$\frac{d}{dx}\left(\frac{2}{\ln(x^2)}\right),$

we can proceed by applying the chain rule and other differentiation techniques.

1. Rewrite the Expression: Note that we can simplify the argument inside the logarithm: $\ln(x^2) = 2\ln(x).$ So the expression becomes: $\frac{2}{\ln(x^2)} = \frac{2}{2\ln(x)} = \frac{1}{\ln(x)}.$

2. Differentiate $\frac{1}{\ln(x)}$:
   Now, let $f(x) = \frac{1}{\ln(x)}$. To find $f'(x)$, we use the quotient rule or the chain rule.

   Rewrite $f(x) = (\ln(x))^{-1}$.

   Using the chain rule: $f'(x) = -(\ln(x))^{-2} \cdot \frac{1}{x} = -\frac{1}{x(\ln(x))^2}.$

Thus, the derivative is

$\frac{d}{dx}\left(\frac{2}{\ln(x^2)}\right) = -\frac{1}{x(\ln(x))^2}.$

Would you like more detailed steps on the differentiation process?

Here are five related questions to expand your understanding:

1. How would you differentiate $\frac{1}{\ln(x^3)}$ with respect to $x$?
2. What is the derivative of $\frac{2}{\ln(x^4)}$?
3. Can you find the derivative of $\frac{3}{\ln(x)}$?
4. How does the chain rule apply in differentiating $\frac{1}{\ln(x^k)}$?
5. What is the importance of rewriting logarithmic expressions before differentiating?

Tip: Simplifying logarithmic expressions before differentiating often makes the process easier and reduces errors.