To avoid "extended discussion", here is what I said.
\begin{align} (f(x).g(x))^{(n)} &=\sum_{k=0}^{n}\binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)\\
&=f(x)g^{(n)}+\sum_{k=1}^{n}\binom{n}{k}f^{(k)}(x)g^{(n-k)}(x) \\
&=\frac2{\sqrt{\pi}}c^ne^{cx}\int_0^{a+x}e^{t^2}dt
+\sum_{k=1}^{n}\binom{n}{k}f^{(k)}(x)g^{(n-k)}(x).
\end{align}
Now use your formulas for $f^{(k)}(x)$, for $k\geq1$, and the obvious one $g^{(k)}(x)$ for any $k$.