What is going on with this generating functional? (QFT)

I am reading Peskin and Schroeder’s chapter on functional methods and they compute the following correlation function:
begin{equation*}
begin{split}
langle 0| Tphi_1phi_2phi_1phi_3 |0rangle
&=
frac{delta}{delta J_1 }
frac{delta}{delta J_2 }
frac{delta}{delta J_3 }[-J_xD_{x4}]e^{-frac{1}{2}J_xD_{xy}J_y}bigg|_{J=0}
&=
frac{delta}{delta J_1 }
frac{delta}{delta J_2 }
[-D_{34}+J_xD_{x4}J_yD_{y3}]
e^{-frac{1}{2}J_xD_{xy}J_y}bigg|_{J=0}
&=
frac{delta}{delta J_1 }
[-D_{34}J_xD_{x2}+D_{24}J_yD_{y3}+J_xD_{x4}D_{23}]
e^{-frac{1}{2}J_xD_{xy}J_y} bigg|_{J=0}
&=D_{34}D_{12}+D_{24}D_{13}+D_{14}D_{23}
end{split}
end{equation*}

However, I don’t understand why in the third line the term
begin{equation*}
J_xD_{x4}J_yD_{y3} frac{delta}{delta J_2} e^{-frac{1}{2}J_xD_{xy}J_y}
=
J_xD_{x4}J_yD_{y3}(-J_zD_{z2})
end{equation*}
doesn’t appear. I would expect this term from the product rule acting on $J_xD_{x4}J_yD_{y3}e^{-frac{1}{2}J_xD_{xy}J_y}$ in the second line.

What am I missing?

Note on notation: I am using Peskin’s short notation where:
$$ -frac{1}{2}J_xD_{xy}J_y=-frac{1}{2} int d^4x’,d^4y’, J(x’) D_F(x’-y’)J(y’) $$