Let $X\subset\mathbb{P}^n$ be a variety with associated homogeneous ideal $I(X) = (f_1,...,f_r)$ where $f_i = f_i(x_0,...,x_n)$ is an homogeneous polynomial, and let $p\in\mathbb{P}^n$ be a point. Here $X$ is not necessarily a complete intersection.
Is it true that $X$ has multiplicity $m$ at $p$ if and only if all the partial derivatives of order less than or equal to $m-1$ of any of the $f_i$'s vanish at $p$ but there is at least a partial derivative of order $m$ of one of the $f_i$'s that does do vanish at $p$ ?
No. For example, look at the variety $C\subseteq\mathbb P^3$ defined by $(z,zw^2-y^3-y^2w+x^2w)$. At the point $(0,0,0,1)$ each of these polynomials has a non vanishing partial derivative of order $1$, but $C$ is a curve with a node (multiplicity $2$) at that point.