$R_{p+1}(x) = \sum_{i_1+i_2+...+i_n = p+1} \frac{1}{i_1!i_2!...i_n!} [\frac{\partial^{p+1}f}{\partial x_1^{i_1} x_2^{i_2}...x_n^{i_n}} (c) × (x_1 - x_{0,1})^{i_1} ...(x_n - x_{0,n})^{i_n}]$
where $c \in (x_0,x)$ and $x = (x_1,x_2,...,x_n)$, $x_0 = (x_{0,1},x_{0,2},...x_{0,n})$
$p=2$ in this case and $x_0 = (0,0,0)$
$R_3(x) = \frac{1}{3!}[\frac{\partial^3f}{\partial x_3^3}(c) × x_3^3] + \frac{1}{3!}[\frac{\partial^3f}{\partial x_2^3}(c) × x_2^3] +\frac{1}{3!}[\frac{\partial^3f}{\partial x_1^3}(c) × x_1^3] + \frac{1}{2!} [\frac{\partial^3f}{\partial x_2 x_3^2}(c) × x_2 x_3^2] + \frac{1}{2!}[\frac{\partial^3f}{\partial x_1 x_3^2}(c) × x_1 x_3^2] + \frac{1}{2!}[\frac{\partial^3f}{\partial x_1 x_2^2}(c) × x_1 x_2^2] + \frac{1}{2!}[\frac{\partial^3f}{\partial x_2^2 x_3}(c) × x_2^2 x_3] + \frac{1}{2!}[\frac{\partial^3f}{\partial x_1^2 x_2}(c) × x_1^2 x_2] + \frac{1}{2!}[\frac{\partial^3f}{\partial x_1^2 x_3}(c) × x_1^2 x_3] + \frac{1}{1!}[\frac{\partial^3f}{\partial x_1 x_2 x_3}(c) × x_1 x_2 x_3]$
Is it correct?
And if I'm given the function: $f(x_1, x_2, x_3) = e^{x_1^2 + x_2 + x_1x_3} cos x_1$,
What is $\frac{\partial^3f}{\partial x_3^3}(c)$? If $\frac{\partial^3f}{\partial x_3^3}(x_1,x_2,x_3) = e^{x_1^2 + x_2 + x_1x_3} x_1^3cosx_1$