Differentiate the following w.r.t.
`(xsinx+cosx)/(xsinx-cosx)`

Answer

To differentiate the function \( y = \frac{x \sin x + \cos x}{x \sin x - \cos x} \) with respect to \( x \), we will use the quotient rule. The quotient rule states that if \( y = \frac{u}{v} \), then the derivative \( \frac{dy}{dx} \) is given by:

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

where \( u = x \sin x + \cos x \) and \( v = x \sin x - \cos x \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = x \sin x + \cos x \)
- \( v = x \sin x - \cos x \)

### Step 2: Differentiate \( u \) and \( v \)
Now we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

**For \( u \):**
Using the product rule on \( x \sin x \):
\[ \frac{du}{dx} = \frac{d}{dx}(x \sin x) + \frac{d}{dx}(\cos x) \]
\[ = (1 \cdot \sin x + x \cdot \cos x) - \sin x \]
\[ = x \cos x \]

**For \( v \):**
Using the product rule on \( x \sin x \):
\[ \frac{dv}{dx} = \frac{d}{dx}(x \sin x) - \frac{d}{dx}(\cos x) \]
\[ = (1 \cdot \sin x + x \cdot \cos x) + \sin x \]
\[ = x \cos x + 2 \sin x \]

### Step 3: Apply the Quotient Rule
Now substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:

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

### Step 4: Simplify the Expression
Now we will simplify the numerator:

1. Expand both terms in the numerator:
- First term: \( (x \sin x - \cos x)(x \cos x) = x^2 \sin x \cos x - x \cos^2 x \)
- Second term: \( (x \sin x + \cos x)(x \cos x + 2 \sin x) = x^2 \sin x \cos x + 2x \sin^2 x + x \cos^2 x + 2 \cos x \sin x \)

2. Combine the terms:
\[ = x^2 \sin x \cos x - x \cos^2 x - (x^2 \sin x \cos x + 2x \sin^2 x + x \cos^2 x + 2 \cos x \sin x) \]
\[ = -2x \sin^2 x - 2 \cos x \sin x \]

Thus, the derivative simplifies to:
\[ \frac{dy}{dx} = \frac{-2(x \sin^2 x + \cos x \sin x)}{(x \sin x - \cos x)^2} \]

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