If `Y = ((2 - 3 cos x)/("sin" x))`, then `(dy)/(dx)` at `x = (pi)/(4)` is

Options

A. `2 (3 - sqrt(2)`
B. `2 - 3 sqrt(2)`
C. `3 sqrt(2) + 1`
D. `6 - sqrt(2)`

Answer

To find \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\) for the function \(Y = \frac{2 - 3 \cos x}{\sin x}\), we will use the quotient rule for differentiation.

### Step-by-Step Solution:

1. **Identify the function**:
\[ Y = \frac{2 - 3 \cos x}{\sin x} \]

2. **Apply the Quotient Rule**:
The quotient rule states that if \(Y = \frac{u}{v}\), then:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
Here, \(u = 2 - 3 \cos x\) and \(v = \sin x\).

3. **Differentiate \(u\) and \(v\)**:
- Differentiate \(u\):
\[ \frac{du}{dx} = 0 + 3 \sin x = 3 \sin x \]
- Differentiate \(v\):
\[ \frac{dv}{dx} = \cos x \]

4. **Substitute into the Quotient Rule**:
\[ \frac{dy}{dx} = \frac{\sin x \cdot (3 \sin x) - (2 - 3 \cos x) \cdot (\cos x)}{\sin^2 x} \]

5. **Simplify the expression**:
\[ = \frac{3 \sin^2 x - (2 \cos x - 3 \cos^2 x)}{\sin^2 x} \]
\[ = \frac{3 \sin^2 x + 3 \cos^2 x - 2 \cos x}{\sin^2 x} \]
Using the identity \(\sin^2 x + \cos^2 x = 1\):
\[ = \frac{3 - 2 \cos x}{\sin^2 x} \]

6. **Evaluate at \(x = \frac{\pi}{4}\)**:
- Calculate \(\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\) and \(\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}\):
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{4}} = \frac{3 - 2 \cdot \frac{1}{\sqrt{2}}}{\left(\frac{1}{\sqrt{2}}\right)^2} \]
\[ = \frac{3 - \frac{2}{\sqrt{2}}}{\frac{1}{2}} = (3 - \sqrt{2}) \cdot 2 = 6 - 2\sqrt{2} \]

### Final Answer:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{4}} = 6 - 2\sqrt{2} \]