  For a bivariate probability distribution, local dependence around a single
point on the support is often formulated as the second derivative of the
logarithm of the probability density function. However, this definition lacks
the invariance under marginal distribution transformations, which is often
required as a criterion for dependence measures. In this study, we examine the
\textit{relative local dependence}, which we define as the ratio of the local
dependence to the probability density function, for copulas. By using this
notion, we point out that typical copulas can be characterised as the solutions
to the corresponding partial differential equations, particularly highlighting
that the relative local dependence of the Frank copula remains constant. The
estimation and visualization of the relative local dependence are demonstrated
using simulation data. Furthermore, we propose a class of copulas where local
dependence is proportional to the $k$-th power of the probability density
function, and as an example, we demonstrate a newly discovered relationship
derived from the density functions of two representative copulas, the Frank
copula and the Farlie-Gumbel-Morgenstern (FGM) copula.
