Multiplicity of Points on Planar Projective Curve

Question

I'm working on Fulton's Algeraic Curves problem 5.6 and I feel I must not understand the question because it seems obviously false. The problem asks to prove that for a curve $F$ in $\mathbb{P}^2$ and a point $P \in F$, show that $m_P(F_X)\geq m_P(F) -1 $. Fulton defines the multiplicity of a point on a projective curve by dehomegonizing and using the multiplicity definition from affine curves. If we take $F = Y^2Z, P = [1:0:1]$ then we have $F_X = 0$, so $m_P(F_X) = m_p((F_X)_*) = 0$, but $m_P(F_*) = 2$ and clearly $0 \not \geq 2 - 1 = 1$. Where am I going wrong?