First, the quantity you defined is the Hilbert-Samuel multiplicity of the ideal $J= (f_x,f_y)$ in $R=\mathbb C[[x,y]]/(f)$. The multiplicity of $f$ usually refers to the multiplicity of the maximal ideal $m$ of $R$.

As you noted, it is enough to show that a generic combinations of the generator of $J$ is a reduction. This is true much more generally, and basically the point is to consider the fiber cone $S= F(I) = k\oplus \frac {J}{mJ}\oplus \frac {J^2}{mJ^2}\oplus... $. This ring has dimension $1$, and is generated in degree one. Then, for a generic linear form $l \in S_1$, $S/lS$ is $0$-dimensional, so $lS_n=S_{n+1}$ for $n\gg 0$. But since $l \in S_1= J/mJ$, this says exactly that $J^nc = J^{n+1}$ where $c$ is a lift of $l$.

First, the quantity you defined is the Hilbert-Samuel multiplicity of the ideal $J= (f_x,f_y)$ in $R=\mathbb C[[x,y]]/(f)$. The multiplicity of $f$ usually refers to the multiplicity of the maximal ideal $m$ of $R$.

As you noted, it is enough to show that a generic combination of the generators of $J$ is a reduction. This is true much more generally, and basically the point is to consider the fiber cone $S= F(I) = k\oplus \frac {J}{mJ}\oplus \frac {J^2}{mJ^2}\oplus... $. This algebra has dimension $1$, and is generated in degree one. Then, for a generic linear form $l \in S_1$, $S/lS$ is $0$-dimensional, so $lS_n=S_{n+1}$ for $n\gg 0$. But since $l \in S_1= J/mJ$, this says exactly that $J^nc = J^{n+1}$ where $c$ is a lift of $l$.