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 the derivative \( \frac{dy}{dx} \) of the function \( Y = \frac{2 - 3 \cos x}{\sin x} \) at \( x = \frac{\pi}{4} \), we will use the quotient rule. The quotient rule states that if you have a function \( \frac{u}{v} \), then the derivative is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Where:
- \( u = 2 - 3 \cos x \)
- \( v = \sin x \)

### Step 1: Differentiate \( u \) and \( v \)

First, we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

1. **Differentiate \( u \)**:
\[ u = 2 - 3 \cos x \implies \frac{du}{dx} = 0 + 3 \sin x = 3 \sin x \]

2. **Differentiate \( v \)**:
\[ v = \sin x \implies \frac{dv}{dx} = \cos x \]

### Step 2: Apply the Quotient Rule

Now we can apply the quotient rule:

\[ \frac{dy}{dx} = \frac{\sin x \cdot (3 \sin x) - (2 - 3 \cos x) \cdot \cos x}{\sin^2 x} \]

### Step 3: Simplify the Expression

Now we simplify the numerator:

\[ \frac{dy}{dx} = \frac{3 \sin^2 x - (2 \cos x - 3 \cos^2 x)}{\sin^2 x} \]

This can be rewritten as:

\[ \frac{dy}{dx} = \frac{3 \sin^2 x - 2 \cos x + 3 \cos^2 x}{\sin^2 x} \]

### Step 4: Substitute \( x = \frac{\pi}{4} \)

Now we need to evaluate \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \):

1. Calculate \( \sin \frac{\pi}{4} \) and \( \cos \frac{\pi}{4} \):
\[ \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \]

2. Substitute these values into the derivative:
\[ \frac{dy}{dx} = \frac{3 \left(\frac{1}{\sqrt{2}}\right)^2 - 2 \left(\frac{1}{\sqrt{2}}\right) + 3 \left(\frac{1}{\sqrt{2}}\right)^2}{\left(\frac{1}{\sqrt{2}}\right)^2} \]

This simplifies to:
\[ = \frac{3 \cdot \frac{1}{2} - 2 \cdot \frac{1}{\sqrt{2}} + 3 \cdot \frac{1}{2}}{\frac{1}{2}} = \frac{\frac{3}{2} - \frac{2}{\sqrt{2}} + \frac{3}{2}}{\frac{1}{2}} \]

Combine the terms in the numerator:
\[ = \frac{3 - \frac{2\sqrt{2}}{2}}{\frac{1}{2}} = \frac{3 - \sqrt{2}}{\frac{1}{2}} = 2(3 - \sqrt{2}) = 6 - 2\sqrt{2} \]

### Final Answer

Thus, the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) is:

\[ \frac{dy}{dx} = 6 - 2\sqrt{2} \]