Differentiation of Determinants

If Δ(x) =

|f1(x) g1(x)|
|f2(x) g2(x)|

its differentiatin Δ'(x) =

|f1'(x) g1'(x)|
|f2(x) g2(x)|

Plus
|f1(x) g1(x)|
|f2'(x) g2'(x)|

This means to differentiate a determinant, we differentiate one row at a time, keeping others unchanged and then add them

If there are two rows R1 and R2

Δ'(x) =

|R1'|
|R2|

Plus
|R1|
|R2'|