Logarithmic derivative

For a capital amount $A (t)$ evolving according to $A (t) = P e^{r t}$ , the instantaneous rate of change is given by the derivative:

\frac{d A}{d t} = r P e^{r t} = r A (t)

This reveals that the relative rate of growth is constant:

\frac{1}{A (t)} \frac{d A}{d t} = r

In finance context, it is more intuitive thinking in terms of different rates $r (t)$ , depending on time. And the accumulated money will be:

A (t) = P \exp (\int_{0}^{t} r (τ) d τ)

Relationship to the Logarithmic Derivative:
The expression $\frac{d}{d t} \ln (A (t))$ is the logarithmic derivative of $A (t)$ , which serves as a fundamental operator for extracting the instantaneous relative growth rate:

\frac{d}{d t} \ln (A (t)) = \frac{A^{'} (t)}{A (t)} = r (t)

In financial terms, this logarithmic derivative isolates the continuous interest rate $r$ , stripping away the scale of the principal $P$ and the accumulated interest $A (t)$ , allowing us to quantify the growth velocity of the investment at any instant $t$ .
Related flow theorem for vector fields#Jacobian of the flow of a vector field.