In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma:
Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying $$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t,$$ and let $f(x,t)$ be twice differentiable function; then we have that $f(X_t,t)$ is an Ito process, and that $$d(f(X_t,t)) = \frac{\partial f}{\partial t}(X_t,t)dX_t + f'(X_t,t)dX + \frac{1}{2}f''(X_t,t) dX_t^2$$ where $dX_t^2$ is defined by $$dt^2 = 0, \quad dtdW_t = 0\quad dW_t^2=dt.$$
I have some doubt on the Ito's Lemma above. Is it stated correctly? What is the meaning of $f'(X_t,t)$ and $f''(X_t,t)$ as we have multivariable function? Also, the Ito's Lemma that I know is defined by $$d(f(X_t,t)) = \frac{\partial f}{\partial t}(X_t,t) dt + \frac{\partial f}{\partial x}(x,t) dX_t + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}(x,t) (dX_t)^2$$ where $(dX_t)^2 = \sigma^2(X_t,t)dt.$
Are the two Ito's Lemma above equivelent?
