Given `s=t^(2)+5t+3`, find `(ds)/(dt)`

Explanation

To solve the problem of finding \(\frac{ds}{dt}\) given the equation \(s = t^2 + 5t + 3\), we will follow these steps:

### Step 1: Write down the equation
We start with the equation given in the problem:
\[ s = t^2 + 5t + 3 \]

### Step 2: Differentiate both sides with respect to \(t\)
To find \(\frac{ds}{dt}\), we differentiate \(s\) with respect to \(t\). We will apply the rules of differentiation to each term on the right side of the equation.

- The derivative of \(t^2\) is \(2t\).
- The derivative of \(5t\) is \(5\).
- The derivative of a constant (which is \(3\)) is \(0\).

So, differentiating both sides gives us:
\[ \frac{ds}{dt} = \frac{d}{dt}(t^2) + \frac{d}{dt}(5t) + \frac{d}{dt}(3) \]

### Step 3: Calculate the derivatives
Now we compute the derivatives:
\[ \frac{ds}{dt} = 2t + 5 + 0 \]

### Step 4: Simplify the expression
Combining the terms, we find:
\[ \frac{ds}{dt} = 2t + 5 \]

### Final Answer
Thus, the final answer is:
\[ \frac{ds}{dt} = 2t + 5 \]

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