Let $E$ be a smooth curve, that is, one dimensional projective variety with dimension one, and Jacobi matrix is non-singular at any point. And suppose that the map $φ$, which sends $E$ to $E'$, is isomorphism. Isomorphism means there are two morphism on both direction which composites identity.
Then, can we say that $E'$ is smooth?
Yes.
An isomorphism $\varphi$ of varieties induces an isomorphism of local rings $\mathcal O_x\cong \mathcal O_{\varphi(x)}$ at every point, hence an isomorphism of the maximal ideals $\mathfrak m_x\cong \mathfrak m_{\varphi(x)}$ and hence an isomorphism of the cotangent spaces $\mathfrak m_x/\mathfrak m^2_x\cong \mathfrak m_{\varphi(x)}/\mathfrak m^2_{\varphi(x)}.$ It follows that the dimension of the cotangent spaces at $x$ and $\varphi(x)$ are the same, so $x$ is a smooth point of $E$ exactly when $\varphi(x)$ is a smooth point of $E'.$