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The given function is:

\[
g(x) = x^{\cos(x)}
\]

To find the derivative \( g'(x) \) of this function, we can use logarithmic differentiation, which is especially useful for functions where the exponent is also a function of \( x \).

### Step-by-step solution:

1. **Take the natural logarithm of both sides**:
   \[
   \ln(g(x)) = \ln(x^{\cos(x)})
   \]

2. **Simplify using logarithm properties**:
   \[
   \ln(g(x)) = \cos(x) \ln(x)
   \]

3. **Differentiate both sides with respect to \( x \)**:
   Use the chain rule on the left:
   \[
   \frac{g'(x)}{g(x)} = \cos(x) \cdot \frac{1}{x} + \ln(x) \cdot (-\sin(x))
   \]

   Simplifying this gives:
   \[
   \frac{g'(x)}{g(x)} = \frac{\cos(x)}{x} - \sin(x) \ln(x)
   \]

4. **Solve for \( g'(x) \)**:
   Multiply both sides by \( g(x) = x^{\cos(x)} \):
   \[
   g'(x) = x^{\cos(x)} \left( \frac{\cos(x)}{x} - \sin(x) \ln(x) \right)
   \]

### Final derivative:
\[
g'(x) = x^{\cos(x)} \left( \frac{\cos(x)}{x} - \sin(x) \ln(x) \right)
\]

Would you like more details or have any specific questions on this?

Here are 5 related questions to deepen understanding:
1. How do you apply logarithmic differentiation in general?
2. What is the domain of the function \( g(x) = x^{\cos(x)} \)?
3. How would you find the critical points of \( g(x) \)?
4. Can \( g(x) \) have any asymptotes?
5. How does \( g(x) \) behave as \( x \to 0^+ \) and \( x \to \infty \)?

**Tip:** Logarithmic differentiation is especially useful when dealing with expressions like \( f(x)^{g(x)} \), where both the base and exponent are functions of \( x \).

Find the derivative of the function g(x) = x^cos(x).

Learn how to find the derivative of the function g(x) = x^cos(x) using logarithmic differentiation. This step-by-step solution covers key calculus concepts, including exponentiation and chain rule, and is suitable for university-level students.

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