Taylor's Theorem

For a basic function $f:\mathbb{R}\to \mathbb{R}$, $f(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)+R(x,x_0)$ and $R(x,x_0)\ll f(x_0{)}^{\prime }(x-x_0)$
This is a first order Taylor expansion

If $f^{\prime \prime }$ is continuous then you can further the expansion $f(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)+\frac{1}{2}{f}^{\prime \prime }(x_0)(x-x_0{)}^2+R^{(2)}(x,x_0)$

For a multivariable function $f:{\mathbb{R}}^2\to \mathbb{R}$,
$f(x)=f(x_0)+\nabla f(x_0)(x-x_0)+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x_{(1)}^2}(x_{(1)}-x_{0(1)}{)}^2+\frac{{\partial }^{2}f}{\partial x_{(1)}{x}_{(2)}}(x_{(1)}-x_{0(1)})(x_{(2)}-x_{0(2)})+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x_{(2)}^2}(x_{(2)}-x_{0(2)}{)}^2+R^{(2)}(x,x_0)$