FieldDerivative[f[x], x, li1, li2, ...] is the derivative of f[x] with respect to space-time variables x and with Lorentz indices li1, li2, ..., where li1, li2, ... have head LorentzIndex.
FieldDerivative[f[x], x, li1, li2, ...] can be given as FieldDerivative[f[x], x, {l1, l2, ...}], where l1 is li1 without the head.
FieldDerivative is defined only for objects with head QuantumField. If the space-time derivative of other objects is wanted, the corresponding rule must be specified.
Overview, FCPartialD, ExpandPartialD.
QuantumField[A, {\[Mu]}][x] . QuantumField[B, {\[Nu]}][y] . QuantumField[C, {\[Rho]}][x] . QuantumField[D, {\[Sigma]}][y]A_{\mu }(x).B_{\nu }(y).C_{\rho }(x).D_{\sigma }(y)
FieldDerivative[%, x, {\[Mu]}] // DotExpandA_{\mu }(x).B_{\nu }(y).\left(\left.(\partial _{\mu }C_{\rho }\right)\right)(x).D_{\sigma }(y)+\left(\left.(\partial _{\mu }A_{\mu }\right)\right)(x).B_{\nu }(y).C_{\rho }(x).D_{\sigma }(y)
FieldDerivative[%%, y, {\[Nu]}] // DotExpandA_{\mu }(x).B_{\nu }(y).C_{\rho }(x).\left(\left.(\partial _{\nu }D_{\sigma }\right)\right)(y)+A_{\mu }(x).\left(\left.(\partial _{\nu }B_{\nu }\right)\right)(y).C_{\rho }(x).D_{\sigma }(y)