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## Main Question or Discussion Point

I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from:

$$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$

To:

$$-\frac{\delta}{\delta J(x_1)} [ -\frac{1}{2} \int d^4 y \; D_F (x_2-y)J(y) - \frac{1}{2} \int d^4 x \; J(x) D_F (x-x_2)]\frac{Z[J]}{Z_0} |_{J=0} \; \; \;\text{?}$$

Why is the functional derivative with respect to ##J(x_2)## equal to the term within the brackets above?

$$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$

To:

$$-\frac{\delta}{\delta J(x_1)} [ -\frac{1}{2} \int d^4 y \; D_F (x_2-y)J(y) - \frac{1}{2} \int d^4 x \; J(x) D_F (x-x_2)]\frac{Z[J]}{Z_0} |_{J=0} \; \; \;\text{?}$$

Why is the functional derivative with respect to ##J(x_2)## equal to the term within the brackets above?