Use the formula for the Riemannian volume form $\sigma = vol_g$ in the coordinates $(x^1,\dots,x^n)$ in $\Bbb R^n$ $ vol_g = \sqrt{|\det{g}|} dx^1 \wedge \dots \wedge dx^n $ where $g$ is the first fundamental form of the surface $X$ which is given by $ g_{i j} \colon = \langle\varphi_i, \varphi_j\rangle $ where $\langle\cdot,\cdot\rangle$ denotes the standard inner product ("dot-product") in $R^{n+1}$, and vectors $\varphi_i$ are defined as $ \varphi_i \colon = \varphi_* (\partial_i) $
Notice, that your $\varphi$ is usually referred to as a parametrization, not as a chart. Using a parametrization we identify the surface with the parameter space (locally).
More explicitly your parametization looks as $ \varphi \colon \Bbb R^n \to \Bbb R^{n+1} \colon \, x \mapsto (x,f(x)) \colon \, (x_1, \dots, x_n) \mapsto (x_1, \dots, x_n, f(x_1, \dots, x_n)) $
In the given parametrization we have $ \varphi_i = \begin{pmatrix} 0 \\ \dots \\ 1 \\ \dots \\ 0 \\ f_i \end{pmatrix} $ where $f_i \colon = \frac{\partial f}{\partial x^i}$
Calculate the first fundamental form and its determinant, and you are done!