Properties between Pearson correlation of two function and the correlations of their derivatives

Question

For two functions, $f(t)$ and $g(t)$, I can calculate the Pearson correlation $Corr$.

Does the correlation, $CorrD$, between the derivative functions, $f'(t)$ and $g'(t)$, has some properties with $Corr$ ?

We can assume f and g are smooth (infinitely derivable)