Why not work in small but certain steps? Simply start without the vector notation:
$$
f(x+\Delta x,y+\Delta y,z + \Delta z) = f(x,y,z)
+ \Delta x \frac{\partial f}{\partial x}
+ \Delta y \frac{\partial f}{\partial y}
+ \Delta z \frac{\partial f}{\partial z}\\
+ \frac{1}{2}(\Delta x)^2 \frac{\partial^2 f}{\partial x^2}
+ \frac{1}{2}(\Delta y)^2 \frac{\partial^2 f}{\partial y^2}
+ \frac{1}{2}(\Delta z)^2 \frac{\partial^2 f}{\partial z^2}\\
+ (\Delta x)(\Delta y) \frac{\partial^2 f}{\partial x \partial y}
+ (\Delta y)(\Delta z) \frac{\partial^2 f}{\partial y \partial z}
+ (\Delta z)(\Delta x) \frac{\partial^2 f}{\partial z \partial x}
$$
Then convert this to the vector notation and check out if you can reproduce the above result:
$$
f(x+\Delta x,y+\Delta y),z + \Delta z) = f(x,y,z) +
\left[ \Large \begin{array}{ccc} \frac{\partial f}{\partial x} &
\frac{\partial f}{\partial y} & \frac{\partial f}{\partial z} \end{array} \right]
\left[ \begin{array}{c} \Delta x \\ \Delta y \\ \Delta z \end{array} \right] \\
+ \frac{1}{2} \left[ \begin{array}{ccc} \Delta x & \Delta y & \Delta z \end{array} \right]
\left[ \Large \begin{array}{ccc} \frac{\partial^2 f}{\partial x^2} &
\frac{\partial^2 f}{\partial x \partial y} &
\frac{\partial^2 f}{\partial x \partial z} \\
\frac{\partial^2 f}{\partial y \partial x} &
\frac{\partial^2 f}{\partial y^2} &
\frac{\partial^2 f}{\partial y \partial z} \\
\frac{\partial^2 f}{\partial z \partial x} &
\frac{\partial^2 f}{\partial z \partial y} &
\frac{\partial^2 f}{\partial z^2} \end{array}\right]
\left[ \begin{array}{c} \Delta x \\ \Delta y \\ \Delta z \end{array} \right]
$$