I am trying to derive the integral of a function where the $x$ term is not a power but a power and squared.
I.e. $$y=n^{x^2}$$ Which similarly would be: $$y =(n^x)^x$$
Similarly, a solution for the derivative would be of interest too.
Hoping there is someone who can shed some light on this for me.
For the integral, you won't get a nice, clean solution. As pointed out in the comments,
$$\int n^{x^2} {\rm d}x = \frac{1}{\sqrt{\ln n}} \int e^{t^2} {\rm d}t$$
using the transformation $t = \sqrt{\ln n}\,x$. Integrals of squared exponentials tend to be messy. For a list of common integrals of this type, have a look at this Wikipedia page. The entry for $\int e^{x^2}{\rm d}x$ is in the Indefinite integrals section.
For the derivative, you can apply the chain rule.
\begin{align} \frac{d}{dx} n^{x^2} &= (\ln n)\,n^{x^2} \frac{d}{dx} x^2 \\ &= (2x \ln n) \, n^{x^2} \end{align}