To find the derivative of \( f(x) = x^{\ln(x)} \), we will use logarithmic differentiation. The steps are as follows:

### Step 1: Take the natural logarithm of both sides.
Taking the natural logarithm on both sides helps simplify the exponent. We get:
\[
\ln(f(x)) = \ln(x^{\ln(x)}) = \ln(x) \cdot \ln(x) = (\ln(x))^2
\]

### Step 2: Differentiate both sides.
Now, differentiate both sides with respect to \( x \).

For the left-hand side:
\[
\frac{d}{dx} \ln(f(x)) = \frac{1}{f(x)} \cdot f'(x)
\]

For the right-hand side, using the chain rule:
\[
\frac{d}{dx} (\ln(x))^2 = 2\ln(x) \cdot \frac{d}{dx} \ln(x) = 2\ln(x) \cdot \frac{1}{x}
\]

### Step 3: Solve for \( f'(x) \).
We now have the equation:
\[
\frac{1}{f(x)} \cdot f'(x) = \frac{2\ln(x)}{x}
\]

Multiply both sides by \( f(x) = x^{\ln(x)} \) to isolate \( f'(x) \):
\[
f'(x) = x^{\ln(x)} \cdot \frac{2\ln(x)}{x}
\]

Simplifying:
\[
f'(x) = x^{\ln(x) - 1} \cdot 2\ln(x)
\]

Thus, the derivative of \( f(x) = x^{\ln(x)} \) is:
\[
f'(x) = 2 \ln(x) \cdot x^{\ln(x) - 1}
\]

Would you like more details or further clarification?

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Here are 5 related questions:
1. How do you find the derivative of \( x^x \)?
2. What is logarithmic differentiation and when is it used?
3. Can you differentiate \( x^{\sin(x)} \)?
4. What are the rules for differentiating functions with variable exponents?
5. How can you apply the chain rule in implicit differentiation?

**Tip:** Logarithmic differentiation is often useful when dealing with variable exponents like \( x^{\ln(x)} \).

Explore the step-by-step solution for finding the derivative of x^(ln(x)) using logarithmic differentiation. This problem involves key calculus concepts such as the chain rule and logarithmic differentiation. Suitable for undergraduate students in calculus courses.

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