If f is a vector function that is written as f ( t ) = f 1 ( t ) ı ı ^ ^ + f 2 ( t ) ȷ ȷ ^ ^ + f 3 ( t ) k ^ ^ , and we want to examine its first derivative then (using the product rule of differentiation): d d t f = d f 1 d t ı ı ^ ^ + d ı ı ^ ^ d t f 1 + d f 2 d t ȷ ȷ ^ ^ + d ȷ ȷ ^ ^ d t f 2 + d f 3 d t k ^ ^ + d k ^ ^ d t f 3 = d f 1 d t ı ı ^ ^ + d f 2 d t ȷ ȷ ^ ^ + d f 3 d t k ^ ^ + [ Ω Ω × × ( f 1 ı ı ^ ^ + f 2 ȷ ȷ ^ ^ + f 3 k ^ ^ ) ] = ( d f d t ) r + Ω Ω × × f where ( d f d t ) r denotes the rate of change of f as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as: d d t f = [ ( d d t ) r + Ω Ω × × ] f .