This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is, | f Y ( y ) d y | = | f X ( x ) d x | , or f Y ( y ) = | d x d y | f X ( x ) = | d d y ( x ) | f X ( x ) = | d d y ( g − − 1 ( y ) ) | f X ( g − − 1 ( y ) ) = | ( g − − 1 ) ′ ( y ) | ⋅ ⋅ f X ( g − − 1 ( y ) ) .