For example in AP: $5,13,21,29 \dots$ the sum of $n$ terms $= 4n^2+n$ and its derivative will be $8n+1$.
The coefficient of $n$ in derivative gives the common difference but what does that '$1$' signify?
And why it is not same as the equation for $n$'th term of AP ?
On
For a progression with general term $ak+b$, the sum for $k$ from $1$ to $n$ is
$$S(n):=n\frac{a(n+1)+2b}2,$$ and its derivative
$$S'(n)=an+\frac a2+b=an+(a+b)-\frac a2.$$
In your case, $8n-3$ yields $8n+1$ and the "$1$" is the first term minus the half common difference, $5-4$.
When the indexes are $0$-based, the derivative of the sum from $0$ to $n$ is
$$S'(n)=\left(n\frac{an+2b}2\right)'=an+b.$$
Sum of $n$ terms of an A.P. with first term $a$ and common difference $d$ is given by : $$S=\frac{n^2d}{2} + n(a-d/2)$$
Therefore it's derivative is :
$$S' = nd + (a-d/2)$$
Therefore coefficient of $n$ represents common difference and constant term is $a-\dfrac{d}{2}$
Note : This function isn't derivable for $n \in \mathbb{N}$, because it is discontinuous.