Find the derivatives of the following functions (1-3) at any point of their domains :
`u=t^(5)/5-t^(3)/3+t`

Answer

To find the derivative of the function \( u = \frac{t^5}{5} - \frac{t^3}{3} + t \), we will differentiate each term with respect to \( t \).

### Step-by-Step Solution:

1. **Identify the function**:
\[ u = \frac{t^5}{5} - \frac{t^3}{3} + t \]

2. **Differentiate each term**:
- The derivative of \( \frac{t^5}{5} \):
\[ \frac{d}{dt}\left(\frac{t^5}{5}\right) = \frac{1}{5} \cdot 5t^{5-1} = t^4 \]
- The derivative of \( -\frac{t^3}{3} \):
\[ \frac{d}{dt}\left(-\frac{t^3}{3}\right) = -\frac{1}{3} \cdot 3t^{3-1} = -t^2 \]
- The derivative of \( t \):
\[ \frac{d}{dt}(t) = 1 \]

3. **Combine the derivatives**:
Now, we combine the derivatives of each term:
\[ u' = t^4 - t^2 + 1 \]

4. **Final result**:
Thus, the derivative of the function \( u \) is:
\[ u' = t^4 - t^2 + 1 \]