Vandermonde Determinant: |1 a a^2| |1 b b^2| = (a-b)(b-c)(c-a) |1 c c^2|

This appears frequently in JEE. Memorize the result AND the sign convention.

Circulant Determinant (a, b, c): |a b c| |c a b| = -(a^3 + b^3 + c^3 - 3abc) = -(a+b+c)(a^2+b^2+c^2-ab-bc-ca) |b c a|

Determinant with arithmetic progression entries: If rows or columns form an AP, often the determinant is zero (because one row becomes a linear combination of others).

Differentiation of determinants: Differentiate one row at a time, keeping others fixed. Sum the resulting determinants. This appears in problems like "If f(x) = |...|, find f'(0)."