Differentiate the following functions w.r.t. x:
`( cos x)/( 5x)`

Answer

To differentiate the function \( y = \frac{\cos x}{5x} \) with respect to \( x \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative is given by:

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

where \( u = \cos x \) and \( v = 5x \).

### Step-by-step Solution:

1. **Identify \( u \) and \( v \)**:
- Let \( u = \cos x \)
- Let \( v = 5x \)

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) with respect to \( x \) is:
\[ \frac{du}{dx} = -\sin x \]
- The derivative of \( v \) with respect to \( x \) is:
\[ \frac{dv}{dx} = 5 \]

3. **Apply the Quotient Rule**:
- According to the quotient rule:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
- Substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = \frac{(5x)(-\sin x) - (\cos x)(5)}{(5x)^2} \]

4. **Simplify the Expression**:
- This simplifies to:
\[ \frac{dy}{dx} = \frac{-5x \sin x - 5 \cos x}{25x^2} \]
- Factor out \(-5\) from the numerator:
\[ \frac{dy}{dx} = \frac{-5(x \sin x + \cos x)}{25x^2} \]
- Simplify further:
\[ \frac{dy}{dx} = \frac{-(x \sin x + \cos x)}{5x^2} \]

5. **Final Result**:
- Therefore, the derivative of \( y = \frac{\cos x}{5x} \) is:
\[ \frac{dy}{dx} = \frac{-(x \sin x + \cos x)}{5x^2} \]