If `s= ut +(1)/(2) at^(2)`, where u and a are constants. Obtain the value of `(ds)/(dt)`.

Explanation

To find the value of \(\frac{ds}{dt}\) from the equation \(s = ut + \frac{1}{2}at^2\), we will differentiate the equation with respect to time \(t\).

### Step-by-Step Solution:

1. **Write down the equation:**
\[ s = ut + \frac{1}{2}at^2 \]

2. **Differentiate both sides with respect to \(t\):**
We will apply the differentiation operator \(\frac{d}{dt}\) to both sides of the equation.
\[ \frac{ds}{dt} = \frac{d}{dt}(ut) + \frac{d}{dt}\left(\frac{1}{2}at^2\right) \]

3. **Differentiate the first term \(ut\):**
Since \(u\) is a constant, the derivative of \(ut\) with respect to \(t\) is:
\[ \frac{d}{dt}(ut) = u \]

4. **Differentiate the second term \(\frac{1}{2}at^2\):**
Here, we apply the power rule of differentiation. The derivative of \(t^2\) is \(2t\), so:
\[ \frac{d}{dt}\left(\frac{1}{2}at^2\right) = \frac{1}{2}a \cdot 2t = at \]

5. **Combine the results:**
Now, we can combine the derivatives from steps 3 and 4:
\[ \frac{ds}{dt} = u + at \]

### Final Answer:
Thus, the value of \(\frac{ds}{dt}\) is:
\[ \frac{ds}{dt} = u + at \]