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You are asked to use logarithmic differentiation to find the derivative of the function:

\[
y = (\cos x)^x
\]

### Steps for Logarithmic Differentiation:

1. **Take the natural logarithm of both sides** to make the exponent easier to handle:
   \[
   \ln y = \ln \left( (\cos x)^x \right)
   \]
   Using the property of logarithms, \(\ln(a^b) = b \ln(a)\), we simplify the right-hand side:
   \[
   \ln y = x \ln(\cos x)
   \]

2. **Differentiate both sides** with respect to \(x\):
   - On the left-hand side, using the chain rule:
     \[
     \frac{d}{dx} \ln y = \frac{1}{y} \frac{dy}{dx}
     \]
   - On the right-hand side, apply the product rule to \(x \ln(\cos x)\):
     \[
     \frac{d}{dx} \left( x \ln(\cos x) \right) = \ln(\cos x) \cdot \frac{d}{dx}(x) + x \cdot \frac{d}{dx}(\ln(\cos x))
     \]
     The derivative of \(x\) is 1, and the derivative of \(\ln(\cos x)\) is:
     \[
     \frac{d}{dx} \ln(\cos x) = \frac{-\sin x}{\cos x} = -\tan x
     \]
     Therefore, the derivative of \(x \ln(\cos x)\) is:
     \[
     \ln(\cos x) - x \tan x
     \]

3. **Combine the results**:
   \[
   \frac{1}{y} \frac{dy}{dx} = \ln(\cos x) - x \tan x
   \]

4. **Solve for \(\frac{dy}{dx}\)** by multiplying both sides by \(y\):
   \[
   \frac{dy}{dx} = y \left( \ln(\cos x) - x \tan x \right)
   \]
   Since \(y = (\cos x)^x\), substitute it back:
   \[
   \frac{dy}{dx} = (\cos x)^x \left( \ln(\cos x) - x \tan x \right)
   \]

This is the derivative of the function \(y = (\cos x)^x\) using logarithmic differentiation.

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Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How does the chain rule apply in logarithmic differentiation?
2. Why do we need logarithmic differentiation for functions like \( (\cos x)^x \)?
3. Can logarithmic differentiation be used for functions with other forms, like \( x^x \)?
4. How does the product rule work in this context of differentiating \( x \ln(\cos x) \)?
5. What are other common situations where logarithmic differentiation simplifies the problem?

### Tip: 
Logarithmic differentiation is particularly useful for functions that involve variables in both the base and the exponent, simplifying otherwise difficult derivatives.

Use logarithmic differentiation to find the derivative of the function y = (cos(x))^x

This page explains how to use logarithmic differentiation to find the derivative of the function y = (cos(x))^x. The solution involves logarithmic properties, the chain rule, and the product rule. Ideal for students in Grades 11-12 or college-level calculus.

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