Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity.

Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$ such that $(\partial_t(F))^m \in \langle \partial_1(F), \dots, \partial_n(F) \rangle$.

Can we conclude that $t$ is a non zero divisor of the quotient ring $\mathbb{C}\{x_1,\dots,x_n,t\}/I$, where

 $I = {\langle \partial_1(F), \dots, \partial_n(F), \partial_t(F) \rangle}$ ?

Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity.

Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$ such that $(\partial_t(F))^m \in \langle \partial_1(F), \dots, \partial_n(F) \rangle$.

Can we conclude that $t$ is a non zero divisor of the quotient ring $\mathbb{C}\{x_1,\dots,x_n,t\}/I$, where  $I = {\langle \partial_1(F), \dots, \partial_n(F), \partial_t(F) \rangle}$ ?