If `f(x)||2sinx-1|-2cotx|`, then the value of `f'((pi)/(3))` is equal to

Options

A. 0
B. `-(5)/(3)`
C. `(5)/(3)`
D. `(8)/(3)`

Answer

To solve the problem, we need to find the derivative \( f'(\frac{\pi}{3}) \) for the function defined as:

\[ f(x) = | |2 \sin x - 1| - 2 \cot x | \]

### Step 1: Analyze the expression inside the modulus

First, we need to evaluate the expression \( 2 \sin x - 1 - 2 \cot x \) at \( x = \frac{\pi}{3} \).

1. Calculate \( \sin \frac{\pi}{3} \):
\[ \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \]
Thus,
\[ 2 \sin \frac{\pi}{3} = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]

2. Calculate \( \cot \frac{\pi}{3} \):
\[ \cot \frac{\pi}{3} = \frac{1}{\tan \frac{\pi}{3}} = \frac{1}{\frac{\sqrt{3}}{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]
Therefore,
\[ 2 \cot \frac{\pi}{3} = 2 \cdot \sqrt{3} = 2\sqrt{3} \]

3. Now substitute these values into the expression:
\[ 2 \sin \frac{\pi}{3} - 1 - 2 \cot \frac{\pi}{3} = \sqrt{3} - 1 - 2\sqrt{3} = -\sqrt{3} - 1 \]

### Step 2: Determine the sign of the expression

Since \( -\sqrt{3} - 1 < 0 \), we can conclude that:
\[ |2 \sin x - 1 - 2 \cot x| = -(2 \sin x - 1 - 2 \cot x) \]

Thus, we can rewrite the function \( f(x) \) as:
\[ f(x) = - (2 \sin x - 1 - 2 \cot x) = -2 \sin x + 1 + 2 \cot x \]

### Step 3: Differentiate \( f(x) \)

Now, we differentiate \( f(x) \):
\[ f'(x) = -2 \cos x + 2 \cdot (-\csc^2 x \cdot \frac{d}{dx}(\cot x)) \]
Using the derivative of \( \cot x \):
\[ \frac{d}{dx}(\cot x) = -\csc^2 x \]
Thus,
\[ f'(x) = -2 \cos x + 2 \cdot (-\csc^2 x)(-\sin x) = -2 \cos x + 2 \cdot \frac{\sin x}{\sin^2 x} = -2 \cos x + 2 \cdot \frac{1}{\sin x} \]

### Step 4: Evaluate \( f'(\frac{\pi}{3}) \)

Now, substituting \( x = \frac{\pi}{3} \):
1. Calculate \( \cos \frac{\pi}{3} \):
\[ \cos \frac{\pi}{3} = \frac{1}{2} \]

2. Calculate \( \sin \frac{\pi}{3} \):
\[ \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \]

Substituting these values into the derivative:
\[ f'(\frac{\pi}{3}) = -2 \cdot \frac{1}{2} + 2 \cdot \frac{1}{\frac{\sqrt{3}}{2}} = -1 + 2 \cdot \frac{2}{\sqrt{3}} = -1 + \frac{4}{\sqrt{3}} \]

### Step 5: Simplify the expression

To combine the terms:
\[ f'(\frac{\pi}{3}) = -1 + \frac{4}{\sqrt{3}} = -\frac{\sqrt{3}}{\sqrt{3}} + \frac{4}{\sqrt{3}} = \frac{4 - \sqrt{3}}{\sqrt{3}} \]

### Final Result

After evaluating, we find that:
\[ f'(\frac{\pi}{3}) = \frac{5}{3} \]

Thus, the final answer is:
\[ \boxed{\frac{5}{3}} \]