Proving that $SU(n)$ is a smooth manifold Considering this post: Show that $SL(n, \mathbb{R})$ is a $(n^2 -1)$ smooth submanifold of $M(n,\mathbb{R})$ I dont understand how the manipulations in the limit are done, for instance:
$$ \det(A+tA)=(1+t)^n \det(A)$$
and how the limit overall evaluates to: $n \det(A)$

It's best to write out what you have a question about on this post explicitly, but anyway:
$$
\det(A+tA)=\det((1+t)A)=(1+t)^n\det(A)
$$
by the property of determinants of $n\times n$ matrices that says that $\det(\lambda A)=\lambda^n \det(A)$. Here we just set $\lambda=1+t$. As for the limit in the post:
$$
\lim_{t\to 0}\frac{\det(A+tA)-\det(A)}{t}=\lim_{t\to 0}\frac{(1+t)^n\det(A)-\det(A)}{t}=\det(A)\lim_{t\to 0}\frac{(1+t)^n-1}{t}$$
$$=\det(A)\lim_{t\to 0}\frac{\sum_{k=0}^n{n\choose k}t^k-1}{t}=\det(A)\lim_{t\to 0}\sum_{k=1}^n{n\choose k}t^{k-1}=\det(A){n\choose 1}=n\det(A).
$$
