If `y= |x|^(| sin x|) ` then the value of ` (dy)/(dx) ` at `x=- (pi)/(4)` is equal to

Options

A. `-((pi)/(4))^(1// sqrt(2)) [(1)/(sqrt(2))""log ""(4)/(pi) + (2 sqrt(2))/(pi)]`
B. `((pi)/(4))^(1// sqrt(2)) [(1)/(sqrt(2))""log ""(4)/(pi) + (2 sqrt(2))/(pi)]`
C. `((pi)/(4))^(1// sqrt(2)) [(1)/(sqrt(2))""log ""(pi)/(4) - (2 sqrt(2))/(pi)]`
D. `((pi)/(4))^(1// sqrt(2)) [(1)/(sqrt(2))""log ""(pi)/(4) + (2 sqrt(2))/(pi)]`

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( y = |x|^{|\sin x|} \) at the point \( x = -\frac{\pi}{4} \), we will follow these steps:

### Step 1: Determine the expression for \( y \) at \( x = -\frac{\pi}{4} \)
Since \( x = -\frac{\pi}{4} \) is negative, we have:
\[ |x| = -x = \frac{\pi}{4} \]
Next, we need to evaluate \( |\sin x| \):
\[ \sin\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \quad \text{(since sine is negative in the fourth quadrant)} \]
Thus,
\[ |\sin\left(-\frac{\pi}{4}\right)| = \frac{1}{\sqrt{2}} \]
Now substituting these into \( y \):
\[ y = \left(\frac{\pi}{4}\right)^{\frac{1}{\sqrt{2}}} \]

### Step 2: Take the natural logarithm of both sides
To differentiate \( y \), we will take the natural logarithm:
\[ \ln y = \ln\left(|x|^{|\sin x|}\right) = |\sin x| \ln |x| \]
Using the properties of logarithms, we can rewrite this as:
\[ \ln y = |\sin x| \ln |x| \]

### Step 3: Differentiate both sides with respect to \( x \)
Using implicit differentiation:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(|\sin x|) \ln |x| + |\sin x| \frac{d}{dx}(\ln |x|) \]

### Step 4: Differentiate \( |\sin x| \) and \( \ln |x| \)
For \( |\sin x| \):
\[ \frac{d}{dx}(|\sin x|) = \begin{cases} \cos x & \text{if } \sin x \geq 0 \\ -\cos x & \text{if } \sin x < 0 \end{cases} \]
At \( x = -\frac{\pi}{4} \), \( \sin x < 0 \), so:
\[ \frac{d}{dx}(|\sin x|) = -\cos\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]

For \( \ln |x| \):
\[ \frac{d}{dx}(\ln |x|) = \frac{1}{x} \]

### Step 5: Substitute back into the derivative expression
Now substituting these derivatives back:
\[ \frac{1}{y} \frac{dy}{dx} = \left(-\frac{1}{\sqrt{2}}\right) \ln\left(\frac{\pi}{4}\right) + \frac{1}{-\frac{\pi}{4}} \cdot |\sin\left(-\frac{\pi}{4}\right)| \]
Substituting \( |\sin(-\frac{\pi}{4})| = \frac{1}{\sqrt{2}} \):
\[ \frac{1}{y} \frac{dy}{dx} = -\frac{1}{\sqrt{2}} \ln\left(\frac{\pi}{4}\right) - \frac{4}{\pi} \cdot \frac{1}{\sqrt{2}} \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Now, multiply both sides by \( y \):
\[ \frac{dy}{dx} = y \left(-\frac{1}{\sqrt{2}} \ln\left(\frac{\pi}{4}\right) - \frac{4}{\pi \sqrt{2}}\right) \]
Substituting \( y = \left(\frac{\pi}{4}\right)^{\frac{1}{\sqrt{2}}} \):
\[ \frac{dy}{dx} = \left(\frac{\pi}{4}\right)^{\frac{1}{\sqrt{2}}} \left(-\frac{1}{\sqrt{2}} \ln\left(\frac{\pi}{4}\right) - \frac{4}{\pi \sqrt{2}}\right) \]

### Final Expression
Thus, the value of \( \frac{dy}{dx} \) at \( x = -\frac{\pi}{4} \) is:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{2}} \left(\frac{\pi}{4}\right)^{\frac{1}{\sqrt{2}}} \left(\ln\left(\frac{\pi}{4}\right) + \frac{4}{\pi}\right) \]