I am trying to prove an exercise in Hartshorne's. Let $k_0$ be a field of characteristic $p>0$, let $k=k_0(t)$, and let $X\subseteq A_k^2$ be a curve defined by $y^2=x^p-t$. Show that every local ring on $X$ is a regular local ring but $X$ is not smooth over $k$.
For the first statement, a local ring on a point $q$ should be the localization of ring $\mathcal{O}_X$, i.e. $(k[x,y]/y^2-x^p+t)_q$. And a maximal ideal correspondes to a irreducible polynomial in $(k[x,y]/y^2-x^p+t)$, thus, it is regular local ring.
For the second statement, notice that $0=d(y^2-x^p+t)=2dy-px^{p-1}dx=2dy$, so $\Omega_{X/k}=0$. But $f$ has relative dimension 1, and we conclude that $f$ is not smooth.
I am not sure whether my argument is right. I hope someone can point out my mistake if there is any.