Suppose the price of a stock is a function of time $p(t)$ and over some period of time $\Delta t = t_1 - t_0$ it changes by some amount $\Delta p = p(t_1)-p(t_0)$.
In class I was told that the natural logarithm represents the fractional change in the stock price, $\frac{\Delta p}{p} = \frac{p(t_1)-p(t_0)}{p(t_0)}$ in the limit as $t_1 \rightarrow t_0$. The justification is that $\frac{d}{dt}ln(p(t))) = \frac{\frac{d}{dt}(p(t))}{p(t)} = \frac{\Delta p}{p}$.
This argument is confusing to me since my understanding is that quantities like $\Delta p$ would actually be represented by $(p(t))' dt$ (a limiting slope times a small step in time) and not the slope itself. This might be an insignificant detail but I've seen these "sensitivity" figures come up in a few contexts and I'd like to know if I'm missing something. Is there a detail I'm missing?