Taylor Polynomial to Find Partial Derivative

From General Multivariable Taylor Series, $f(x)\sim {\sum }_{\alpha =1}^k\frac{\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}(x_0)}{\alpha !}{x}^{\alpha }$ Then, for a given partial derivative, $\frac{{\partial }^{n}f}{\partial x^n}$, we can locate it from the given vector $x_0$, we know that: $\frac{\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}(x_0)}{a!}{x}^a=\_$, so we use algebra to find the original partial derivative