# Question #19089

Nov 23, 2017

In reality you can only write an approximate equation $y \left(t + \Delta t\right) \approx y \left(t\right) + y ' \left(t\right) \cdot \Delta t$. If $\mathrm{dt}$ represents an "infinitesimal " amount of time (even though there are no such numbers in the standard real number system), this is then sometimes represented as an "exact" equation $y \left(t + \mathrm{dt}\right) = y \left(t\right) + y ' \left(t\right) \mathrm{dt}$. If you let $\mathrm{dy} = y \left(t + \mathrm{dt}\right) - y \left(t\right)$, then you can rearrange this to say $\mathrm{dy} = y ' \left(t\right) \mathrm{dt}$, which is the "infinitesimal version" of the equation $\frac{\mathrm{dy}}{\mathrm{dt}} = y ' \left(t\right)$.

#### Explanation:

To derive this in the way you want, you can write:

$y \left(t + \Delta t\right) = y \left(t + \Delta t\right) - y \left(t\right) + y \left(t\right)$

$= \frac{y \left(t + \Delta t\right) - y \left(t\right)}{\Delta t} \cdot \Delta t + y \left(t\right)$

Now let $\Delta t \to 0$ and use the definition of the derivative to say that $\frac{y \left(t + \Delta t\right) - y \left(t\right)}{\Delta t} \to y ' \left(t\right)$ as $\Delta t \to 0$ to write $y \left(t + \Delta t\right) \approx y \left(t\right) + y ' \left(t\right) \cdot \Delta t$ (this is an approximation because we did not replace the other $\Delta t$'s with 0's).

This approximation is "good" when $\Delta t$ is "small". The approximation gets "better and better" as $\Delta t$ continues to get "smaller and smaller".

These ideas can be made rigorous. If you are interested in how they can be made rigorous (without using infinitesimals), you can watch my video lectures on introductory Real Analysis .