For a 2x2 determinant: If D(x) = |a(x) b(x)| |c(x) d(x)|

Then D'(x) = |a'(x) b'(x)| + |a(x) b(x)| |c(x) d(x) | |c'(x) d'(x)|

Rule: Differentiate one row at a time, keeping the other row unchanged, and add the results.

For a 3x3 determinant with 3 rows: D'(x) = (differentiate Row 1, keep Rows 2,3) + (keep Row 1, differentiate Row 2, keep Row 3) + (keep Rows 1,2, differentiate Row 3)

JEE Application: Given D(x) as a determinant with polynomial entries, find D'(0) or prove D'(x) has a specific form.

Example: D(x) = |x sin x cos x | |-1 0 1 | |0 1 0 |

D'(x) = |1 cos x -sin x| + |x sin x cos x| + |x sin x cos x| |-1 0 1 | |0 0 0 | |0 0 0 | |0 1 0 | |0 1 0 | |0 0 0 |

The second and third determinants are 0 (rows of zeros). So D'(x) = first determinant only.