Let Z_i (i 1) be a sequence of independent and identically distributed random variables with standard exponential distribution H and Z(n) (n 1) be the corresponding sequence of exponential records associated with Z_i (i 1). Let us call the sequence Z(n) (n 1) the first “record derivative” of the sequence Z_i (i 1). It is known that ν₁ = Z(1), ν₂ = Z(2) - Z(1), . . . are independent variables with distribution H. Let T (n) (n 1) be record times obtained from the sequence ν₁, ν₂, . . . and Y (n) = Z(T (n)),W(n) = Y (n) - Y (n - 1) (n 1). Let us call the sequence Y (n) (n 1) (the main objective of the research of the present paper) the second “record derivative” of the sequence Z_i (i 1). In the present paper, we find the distributions of T (n), Y (n),W(n) and study the Laplace transform of Y (n). A limit result for the sequence Y (n) (n 1) is obtained in the paper. We also propose some methods of generation of T (n) and Y (n).