Growth rate vs. Absolute correlations Conceptually, what is the reason why the correlation between growth rates of A and B would be different from correlation between actual A and B? Under what circumstances would the growth rate correlation be higher? How about lower? How about equal? 
 A: Maybe this is easy to see if you look at deterministic sequences. The growth rate is given by
$$ 100 \cdot \frac{a_{t+1}-a_t}{a_t} $$
and so if you change the sequence by adding a constant to all terms, the numerator is unchanged but not the denominator. So by adding different constants we can change the growth rates, thus the correlation between growth rates, without changing correlation between the series itself.
An example, computed in R:
 cor(1:5,  101:105)
[1] 1
 cor(gr(1:5),  gr(101:105))
[1] 0.9318379
 cor(gr(1:5),  gr(1001:1005))
[1] 0.9290624

where the growth rate is calculated by
 gr <-  
function(x) { 
    n <- length(x)
    (diff(x) / x[1:(n-1)])* 100  
}

(and, in R, 1:5 means the sequence 1, 2, 3, 4, 5).
In all cases here, the absolute sequences has correlation 1, while the correlation of growth rates depends on the starting point.
A: A general definition of growth rate would include the case in which shrinking is occurring. As it turns out, one can generalize the definition of half-life to include negative values, which correspond to growth rather than halving. From the geometry of the tangent to the logarithm of any function, as below for drug concentration and in general for any $f(t)$

one can define half-life as a generalization of halving-time for any function at any instant of time using the limits definition of what a derivative is as $$t_{1/2};f(t)=-\ln(2)\dfrac{1}{\dfrac{d\ln[f(t)]}{dt}}=-\ln(2)\dfrac{f(t)}{f'(t)}\;,$$
which is Eq. (6) of Comparison of the gamma-Pareto convolution with.... Finally, note that if one wishes to define the double-life as positive, then one can change the sign above to be positive
$$t_{2};f(t)=\ln(2)\dfrac{f(t)}{f'(t)}\;,$$
in which case half-lives are then negative values. However, no matter how one defines it, via growth, or shrinking, the only way to do either properly is to allow for when the opposite occurs, that is, if one defines doubling-rate as positive, then halving-rate becomes negative.
Finally, there is no reason for an arbitrary vector $\vec{t}$, and arbitrary functions $f$ and $g$ to expect that
$$r\left[\dfrac{f(\vec{t})}{f'(\vec{t})},\dfrac{g(\vec{t})}{g'(\vec{t})}\right]=r[f(\vec{t}),g(\vec{t})]$$
and I don't think that there is any simple answer to the question as posed without specifying what vectors are being considered.
