Differentiate `(sinx)/(x)` with respect to x.

Answer

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

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

where \( u = \sin x \) and \( v = x \).

### Step-by-Step Solution:

1. **Identify \( u \) and \( v \)**:
- Let \( u = \sin x \)
- Let \( v = x \)

2. **Differentiate \( u \) and \( v \)**:
- \( \frac{du}{dx} = \cos x \) (the derivative of \( \sin x \))
- \( \frac{dv}{dx} = 1 \) (the derivative of \( x \))

3. **Apply the Quotient Rule**:
- Substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:
\[ \frac{dy}{dx} = \frac{x \cdot \cos x - \sin x \cdot 1}{x^2} \]

4. **Simplify the expression**:
- This simplifies to:
\[ \frac{dy}{dx} = \frac{x \cos x - \sin x}{x^2} \]

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