Here is the link: Hessian
I understand everything but this line: $$g(x_0 + \Delta x) − g(x_0) = \frac{dg}{dx} (ξ) \Delta x$$ i.e., $$S (X_0, \Delta x, \Delta y) = \frac{∂φ}{∂x} (ξ, y_0 + \Delta y) − \frac{∂φ}{∂x} (ξ, y_0)\Delta x$$
$\textbf{Question:}$ Can someone explain how this relation is reached?
In the link you provided, if you look up a few lines, you'll see $g(x)$ defined as $$g(x)=\phi(x,y_{0}+\Delta y)-\phi(x,y_{0})$$
So to get to the final relation, they just took the partial derivative with respect to $x$, and evaluated at $x=\xi$.
$$S (X_0, \Delta x, \Delta y)=g(x_0 + \Delta x) − g(x_0) = \frac{d(\phi(x,y_{0}+\Delta y)-\phi(x,y_{0}))}{dx}\bigg|_{x=\xi} \Delta x=\left(\frac{\partial\phi}{\partial x} (ξ, y_0 + \Delta y) − \frac{\partial \phi}{\partial x} (ξ, y_0)\right)\Delta x$$