` y=log ((1-sin x )/(1+sin x )),then (dy)/(dx) =`

Options

A. ` -2sec x `
B. ` 2sec x `
C. `- 2tan x `
D. ` 2tan x `

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( y = \log\left(\frac{1 - \sin x}{1 + \sin x}\right) \), we will follow these steps:

### Step 1: Apply the Chain Rule
We start by using the chain rule for the logarithmic function. The derivative of \( \log(u) \) is given by \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = \frac{1 - \sin x}{1 + \sin x} \).

\[ \frac{dy}{dx} = \frac{1}{\frac{1 - \sin x}{1 + \sin x}} \cdot \frac{d}{dx}\left(\frac{1 - \sin x}{1 + \sin x}\right) \]

### Step 2: Differentiate the Quotient
Next, we need to differentiate \( u = \frac{1 - \sin x}{1 + \sin x} \) using the quotient rule. The quotient rule states:

\[ \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2} \]

Here, \( f(x) = 1 - \sin x \) and \( g(x) = 1 + \sin x \).

- \( f'(x) = -\cos x \)
- \( g'(x) = \cos x \)

Now applying the quotient rule:

\[ \frac{d}{dx}\left(\frac{1 - \sin x}{1 + \sin x}\right) = \frac{(1 + \sin x)(-\cos x) - (1 - \sin x)(\cos x)}{(1 + \sin x)^2} \]

### Step 3: Simplify the Numerator
Now let's simplify the numerator:

\[ = \frac{-\cos x(1 + \sin x) - \cos x(1 - \sin x)}{(1 + \sin x)^2} \]

Combining the terms in the numerator:

\[ = \frac{-\cos x - \sin x \cos x - \cos x + \sin x \cos x}{(1 + \sin x)^2} \]

The \( \sin x \cos x \) terms cancel out:

\[ = \frac{-2\cos x}{(1 + \sin x)^2} \]

### Step 4: Substitute Back into the Derivative
Now substitute this back into our expression for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{1 + \sin x}{1 - \sin x} \cdot \frac{-2\cos x}{(1 + \sin x)^2} \]

### Step 5: Simplify the Expression
Now we can simplify:

\[ \frac{dy}{dx} = \frac{-2\cos x}{(1 - \sin x)(1 + \sin x)} = \frac{-2\cos x}{1 - \sin^2 x} = \frac{-2\cos x}{\cos^2 x} \]

### Step 6: Final Simplification
This simplifies to:

\[ \frac{dy}{dx} = -2 \tan x \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = -2 \tan x \]

---