The Taylor expansion of the function $f(x,y)$ is:
\begin{equation} f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)}{\partial x \partial y} \end{equation}
When $f=(x,y,z)$ is the following true?
$$\begin{align} f(x+u,y+v,z+w) \approx f(x,y,z) &+ u \frac{\partial f (x,y,z)}{\partial x}+v \frac{\partial f (x,y,z)}{\partial y} + w \frac{\partial f (x,y,z)}{\partial z} \\ &+uv \frac{\partial^2 f (x,y,z)}{\partial x \partial y} + vw \frac{\partial^2 f (x,y,z)}{\partial y \partial z}+ uw \frac{\partial^2 f (x,y,z)}{\partial x \partial z} \\ &+ uvw \frac{\partial^3 f (x,y,z)}{\partial x \partial y \partial z} \end{align}$$
The general formula for the Taylor expansion of a sufficiently smooth real valued function $f:\mathbb{R}^n \to \mathbb{R}$ at $\mathbf{x}_0$ is
$$f({\bf{x}})=f({\bf{x}}_0)+\nabla f({\bf{x}}_0) \cdot ({\bf{x}}-{\bf{x}}_0) + \frac{1}{2} ({\bf{x}}-{\bf{x}}_0) \cdot \nabla \nabla f ({\bf{x}}_0) \cdot ({\bf{x}}-{\bf{x}}_0) + O(\lVert\mathbf{{\bf{x}}-{\bf{x}}_0}\rVert^2)$$
If you call ${\bf{x}}-{\bf{x}}_0:={\bf{h}}$ then the above formula can be rewritten as
$$f({\bf{x}}_0+{\bf{h}})=f({\bf{x}}_0)+\nabla f({\bf{x}}_0) \cdot {\bf{h}} + \frac{1}{2} {\bf{h}} \cdot \nabla \nabla f ({\bf{x}}_0) \cdot {\bf{h}} + O(\lVert\mathbf{h}\rVert^2)$$
In these formulas, $\nabla f$ is the (first) gradient of $f$, $\nabla\nabla f$ is usually called the Hessian (second gradient) of $f$, and $O$ is the famous big O notation. You can extend this formulation for functions like $f:\mathbb{R}^n \to \mathbb{R}^m$. You may also find it useful to take a look at this link.