- #1
fog37
- 1,568
- 108
- TL;DR Summary
- Understand derivatives as the limit of a quotient
Hello,
The derivative ##\frac {dy} {dx}## "appears" at first glance to be just the ratio of two infinitesimal quantities ##dy## and ##dx##. However, infinitesimals are not really very very small numbers even if sometimes it is useful and practical to think about them as such. Infinitesimal are more like symbols than numbers since they don't important respect rules that numbers obey (Archimedean rule, etc.)
Numerically, a derivative is approximated and calculated as the ratio of two finite difference ##\Delta y## and ##\Delta x##: $$\frac {\Delta y}{\Delta x}$$ For example, instantaneous speed is numerically calculated by forcing the time interval ##\Delta t =t_2 - t_1## to be finite and very very small. The space interval ##\Delta x =x_2 - x_1## does not have to be necessarily small: the object may undergo a large change in position over that short time interval hence having a large speed. So ##\Delta x =x_2 - x_1## does only need to be finite but not very very small at all.
$$v= \frac {\Delta x}{\Delta t}$$ No problem with that. I read that the actual theoretical derivative ##\frac {dx}{dt}## is NOT the quotient of the limit as ##\Delta t## and ##\Delta x## shrinks to zero but the limit of of the ratio ##\frac {\Delta y}{\Delta x} ##as ##\Delta t \rightarrow 0##...
What is the difference between the concept of quotient of the limit of ##\Delta x## and ##\Delta t## versus the limit of the quotient ? Why is the derivative defined as the limit of the quotient?
Thanks!
The derivative ##\frac {dy} {dx}## "appears" at first glance to be just the ratio of two infinitesimal quantities ##dy## and ##dx##. However, infinitesimals are not really very very small numbers even if sometimes it is useful and practical to think about them as such. Infinitesimal are more like symbols than numbers since they don't important respect rules that numbers obey (Archimedean rule, etc.)
Numerically, a derivative is approximated and calculated as the ratio of two finite difference ##\Delta y## and ##\Delta x##: $$\frac {\Delta y}{\Delta x}$$ For example, instantaneous speed is numerically calculated by forcing the time interval ##\Delta t =t_2 - t_1## to be finite and very very small. The space interval ##\Delta x =x_2 - x_1## does not have to be necessarily small: the object may undergo a large change in position over that short time interval hence having a large speed. So ##\Delta x =x_2 - x_1## does only need to be finite but not very very small at all.
$$v= \frac {\Delta x}{\Delta t}$$ No problem with that. I read that the actual theoretical derivative ##\frac {dx}{dt}## is NOT the quotient of the limit as ##\Delta t## and ##\Delta x## shrinks to zero but the limit of of the ratio ##\frac {\Delta y}{\Delta x} ##as ##\Delta t \rightarrow 0##...
What is the difference between the concept of quotient of the limit of ##\Delta x## and ##\Delta t## versus the limit of the quotient ? Why is the derivative defined as the limit of the quotient?
Thanks!