If derivative of `y=|x-1|^(sinx)` at `x=-(pi)/2` is `(1+api)^(b)` then value of `|1/a+4b|` is

Explanation

To solve the problem, we need to find the derivative of the function \( y = |x - 1|^{\sin x} \) at \( x = -\frac{\pi}{2} \) and express it in the form \( (1 + a\pi)^b \). Then, we will calculate \( | \frac{1}{a} + 4b | \).

### Step 1: Rewrite the function
Since \( x = -\frac{\pi}{2} \) is less than 1, we have:
\[ |x - 1| = 1 - x \]
Thus, we can rewrite the function as:
\[ y = (1 - x)^{\sin x} \]

### Step 2: Take the natural logarithm
Taking the natural logarithm of both sides:
\[ \ln y = \sin x \cdot \ln(1 - x) \]

### Step 3: Differentiate both sides
Using implicit differentiation:
\[ \frac{1}{y} \frac{dy}{dx} = \cos x \cdot \ln(1 - x) + \sin x \cdot \left(-\frac{1}{1 - x}\right) \cdot \frac{d}{dx}(1 - x) \]
Since \(\frac{d}{dx}(1 - x) = -1\), we have:
\[ \frac{1}{y} \frac{dy}{dx} = \cos x \cdot \ln(1 - x) + \frac{\sin x}{1 - x} \]
Thus,
\[ \frac{dy}{dx} = y \left( \cos x \cdot \ln(1 - x) - \frac{\sin x}{1 - x} \right) \]

### Step 4: Substitute \( x = -\frac{\pi}{2} \)
Now we need to evaluate \( \frac{dy}{dx} \) at \( x = -\frac{\pi}{2} \):
1. Calculate \( y \) at \( x = -\frac{\pi}{2} \):
\[ y = (1 - (-\frac{\pi}{2}))^{\sin(-\frac{\pi}{2})} = (1 + \frac{\pi}{2})^{-1} = \frac{1}{1 + \frac{\pi}{2}} \]
2. Calculate \( \cos(-\frac{\pi}{2}) \) and \( \sin(-\frac{\pi}{2}) \):
\[ \cos(-\frac{\pi}{2}) = 0, \quad \sin(-\frac{\pi}{2}) = -1 \]
3. Substitute into the derivative:
\[ \frac{dy}{dx} = \frac{1}{1 + \frac{\pi}{2}} \left( 0 \cdot \ln(1 + \frac{\pi}{2}) - \frac{-1}{1 + \frac{\pi}{2}} \right) \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{1}{1 + \frac{\pi}{2}} \cdot \frac{1}{1 + \frac{\pi}{2}} = \frac{1}{(1 + \frac{\pi}{2})^2} \]

### Step 5: Set the derivative equal to the given form
We have:
\[ \frac{dy}{dx} = \frac{1}{(1 + \frac{\pi}{2})^2} \]
This can be expressed as:
\[ \frac{1}{(1 + \frac{\pi}{2})^2} = (1 + a\pi)^b \]
From comparing, we can see that \( a = \frac{1}{2} \) and \( b = -2 \).

### Step 6: Calculate \( | \frac{1}{a} + 4b | \)
Now we compute:
\[ \frac{1}{a} = \frac{1}{\frac{1}{2}} = 2 \]
\[ 4b = 4 \cdot (-2) = -8 \]
Thus:
\[ \frac{1}{a} + 4b = 2 - 8 = -6 \]
Finally, we take the absolute value:
\[ | \frac{1}{a} + 4b | = |-6| = 6 \]

### Final Answer
The value of \( | \frac{1}{a} + 4b | \) is \( 6 \).