Differentiate the following w.r.t. x or t or u as the case may be:
10. `z= (u)/( u^(2) +1) `

Answer

To differentiate the function \( z = \frac{u}{u^2 + 1} \) with respect to \( u \), 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 is given by:

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

where \( u \) is the numerator and \( v \) is the denominator.

### Step-by-Step Solution:

1. **Identify \( u \) and \( v \)**:
- Here, let \( u = u \) (the numerator) and \( v = u^2 + 1 \) (the denominator).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) with respect to \( u \) is:
\[ \frac{du}{du} = 1 \]
- The derivative of \( v \) with respect to \( u \) is:
\[ \frac{dv}{du} = \frac{d}{du}(u^2 + 1) = 2u \]

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

4. **Simplify the Expression**:
- Simplifying the numerator:
\[ = \frac{u^2 + 1 - 2u^2}{(u^2 + 1)^2} \]
- Combine like terms:
\[ = \frac{-u^2 + 1}{(u^2 + 1)^2} \]

5. **Final Result**:
- Thus, the derivative of \( z \) with respect to \( u \) is:
\[ \frac{dz}{du} = \frac{1 - u^2}{(u^2 + 1)^2} \]