Definition.
For a sequence of $\text{i.i.d.}$ random variables $ \left\{X_{k},k\ge 1 \right \},$called inter-renewal times, let $S_{n}=\sum_{k=1}^{n}X_{k},n\ge 0,$ be the time instant of the $n$th event, and $N(t)=\text{max} \left\{n: S_n \le t \right\}$ be the total number of replacements occurred by time $t$.Then $\left\{S_n \ge 0 \right\}$ is called a renewal process and $\left\{N(t),t\le 0 \right\}$ is called a renewal counting process.
Let the cumulative distribution functions for $S_{n},n\ge 0,$ as follows: $F_{0}(x)=\begin{cases} &0 \quad x<0\\ &1 \quad x\ge0 \end{cases};F_{1}=F(x);F_{n}(x)=F^{n*}(x),$ for $n\ge 1,$ where $F^{n*}(x)$ is the $n$-fold convolution of $F$ with itself.
Next, we can obtain the mean of $N(t)$, which is called the renewal function and denoted by $M(t)=\mathbf{E}[N(t)]$, in terms of $F_n.$ $$M(t)=\mathbf{E}[N(t)]=\sum^{\infty}_{n=1}P(N(t)\ge n)=\sum^{\infty}_{n=1}P(S_{n}\le t)=\sum^{\infty}_{n=1}F_{n}(t).(\star)$$
For the case where the inter-renewal times $X_{1},X_{2},...$ are $\text{i.i.d.}$ continuous random variables(a random variable is called continuous iff its CDF is a continuous function). with $\text{p.d.f.} f(x),$ the renewal function is differentiable and $$m(t)=\frac{\mathrm{d}M(t) }{\mathrm{d} x}=\sum_{n=1}^{\infty}f_{n}(t),$$ where $f_{n}(t)$ is $\text{p.d.f.}$ for $S_{n}.$
I don't know why the renewal function $M(t)$ is differentiable under $\left\{ X_{n}\right\}_{n\ge 1}$ are continuous random variables with $\text{p.d.f.}$ I think we should introduce additional conditions that enable the interchange of the summation and differentiation symbols,such as,$$\forall t\ge 0,F(t)<1, \text{since this implies} $$$$M(t)\le F(t)(1-F(t))^{-1}\text {from } (\star)$$
When the $X_k$s have a density $f$, so that each $S_n$ has density $f_n$, then $F_n(t) =\int_0^t f_n(u)\,du$ for each $t\ge0$ and each $n\ge 1$. Therefore $$ \sum_{n=1}^\infty F_n(t) = \sum_{n=1}^\infty\int_0^t f_n(u)\,du=\sum_{n=1}^\infty\int_0^t f_n(u)\,du=\int_0^t\left(\sum_{n=1}^\infty f_n(u)\right)\,du, $$ the exchange of summation and integration being justified by Tonelli's Theorem. Defining $m(u):=\sum_{n=1}^\infty f_n(u)$, this shows that $$ M(t) =\int_0^t m(u)\,du $$ for all $t\ge 0$. The Lebesgue Differentiation Theorem implies that $M'(t) =m(t)$ for a.e. $t\ge 0$.