To solve the given problem:

We are asked to find the derivative of the expression $2x \log(x^3)$.

Step 1: Simplify the expression.

We can use the logarithmic property: $\log(a^b) = b \log(a)$.

So, $\log(x^3) = 3 \log(x)$.

Thus, the expression becomes: $2x \log(x^3) = 2x \cdot 3 \log(x) = 6x \log(x)$

Step 2: Differentiate the expression.

Now, we need to find the derivative of $6x \log(x)$ with respect to $x$. We apply the product rule here.

The product rule states: $\frac{d}{dx} [uv] = u'v + uv'$ where $u = 6x$ and $v = \log(x)$.

• Derivative of $u = 6x$ is $u' = 6$.
• Derivative of $v = \log(x)$ is $v' = \frac{1}{x}$.

Applying the product rule: $\frac{d}{dx} [6x \log(x)] = 6 \log(x) + 6x \cdot \frac{1}{x}$ Simplifying: $= 6 \log(x) + 6$

Final Answer:

$\frac{d}{dx} [2x \log(x^3)] = 6 \log(x) + 6$

Would you like more details or have any other questions?

Here are some related questions you might find helpful:

1. What is the product rule for differentiation, and when is it used?
2. How do logarithmic differentiation techniques help in solving complex problems?
3. Can we apply chain rule and product rule together in other contexts?
4. How would the derivative change if the base of the logarithm was different (e.g., base 10)?
5. What happens when differentiating logarithms involving powers of variables?

Tip: Always look for opportunities to simplify the logarithmic expressions before applying differentiation rules!