If `y=|sin X^(|x|)` then what is the value of `(dy)/(dx) at x = (pi)/(6)` ?

Options

A. `(2^(pi)/(6))(6 l n 2-sqrt(3 pi))/(6)`
B. `(2^(pi)/(6))(6 l n 2+sqrt(3 pi))/(6)`
C. `(2^(pi)/(6))(6 l n 2+sqrt(3 pi))/(6)`
D. `(2^(pi)/(6))(6 l n 2-sqrt(3 pi))/(6)`

Explanation

To find the derivative of the function \( y = |\sin(x^{|x|})| \) at \( x = \frac{\pi}{6} \), we will follow these steps:

### Step 1: Understand the function
The function involves the absolute value of \( \sin(x^{|x|}) \). First, we need to evaluate \( |x| \) at \( x = \frac{\pi}{6} \).

### Step 2: Evaluate \( |x| \)
Since \( x = \frac{\pi}{6} \) is positive, we have:
\[ |x| = \frac{\pi}{6} \]

### Step 3: Substitute into the function
Now, substitute \( |x| \) into the function:
\[ y = |\sin(x^{|x|})| = |\sin\left(\left(\frac{\pi}{6}\right)^{\frac{\pi}{6}}\right)| \]

### Step 4: Differentiate using the chain rule
To differentiate \( y \), we will use the chain rule. The derivative of \( y \) with respect to \( x \) is given by:
\[ \frac{dy}{dx} = \frac{d}{dx} |\sin(x^{|x|})| = \text{sgn}(\sin(x^{|x|})) \cdot \cos(x^{|x|}) \cdot \frac{d}{dx}(x^{|x|}) \]
where \( \text{sgn} \) is the sign function.

### Step 5: Differentiate \( x^{|x|} \)
To differentiate \( x^{|x|} \), we can use logarithmic differentiation:
\[ \frac{d}{dx}(x^{|x|}) = |x| \cdot x^{|x| - 1} + x^{|x|} \cdot \ln(x) \cdot \frac{d}{dx}(|x|) \]
Since \( |x| = x \) for \( x > 0 \):
\[ \frac{d}{dx}(x^{|x|}) = x \cdot x^{x - 1} + x^{x} \cdot \ln(x) \cdot 1 = x^x + x^x \ln(x) \]

### Step 6: Evaluate at \( x = \frac{\pi}{6} \)
Now we need to evaluate \( \frac{dy}{dx} \) at \( x = \frac{\pi}{6} \):
1. Calculate \( \sin\left(\left(\frac{\pi}{6}\right)^{\frac{\pi}{6}}\right) \).
2. Calculate \( \cos\left(\left(\frac{\pi}{6}\right)^{\frac{\pi}{6}}\right) \).
3. Substitute \( x = \frac{\pi}{6} \) into the derivative expression.

### Step 7: Final Calculation
Substituting all values back into the derivative formula, we will find \( \frac{dy}{dx} \) at \( x = \frac{\pi}{6} \).

### Summary of Steps:
1. Evaluate \( |x| \) at \( x = \frac{\pi}{6} \).
2. Substitute into the function to find \( y \).
3. Differentiate using the chain rule.
4. Differentiate \( x^{|x|} \) using logarithmic differentiation.
5. Evaluate the derivative at \( x = \frac{\pi}{6} \).