How can we define the rate of change of a function at a single point?

Question

I am studying derivatives and the basic idea of derivatives is to define the rate of change of a function. Now change in mathematics occurs in between two points so how can we even define the rate of change at a single point. It's completely invalid concept. Either derivatives are not rate of change or this is a wrong way to get rate of change.

Are you okay with speedometer on your car/bike? It actually shows the rate of change of distance at a point. — AgentS, Nov 12 '19 at 15:52
Rate of change is a value of y changed per value of x. Like in linear function y = 2x , it means that in every value change of x with 1, there will be change in y with 2. If change in x gets 0 i.e. on a single point, the change in y will be 0. As there cant be change on a single point. But in derivatives we take change in y with respect to a single point. Which is not possible. — Ritanshu Singh, Nov 12 '19 at 15:58
Typically the "basic idea" of something is merely a motivation or explanation for the definition. The derivative has a precise definition in terms of limits, have you been taught that? And maybe don't be so arrogant as to call a concept invalid just because you don't understand it... — 79037662, Nov 12 '19 at 16:03
It is defined in a very precise way. It depends on the behaviour of the function near the point, just as temperature, speed, blood pressure, etc. do. You can have a tangent to a surface at a particular point. — copper.hat, Nov 12 '19 at 16:16

score 1 · Answer 1 · answered Nov 12 '19 at 15:57

1

In fact, if you had just a point, you could not take the derivative !

It is the existence and continuity of a given function around that point that give sense to the derivative (of that function, at that point).

answered Nov 12 '19 at 15:57

G Cab

35,272

If that's the case then derivative can't be a tangent as tangent is a slope of single point. – Ritanshu Singh Nov 12 '19 at 16:00

score 0 · Answer 2 · edited Nov 12 '19 at 16:34

0

The rate/slope determined by differentiation is in fact between two points $x$ and $x+dx$. $$f'(x)=\lim_{dx\to 0}{f(x+dx)-f(x)\over (x+dx)-(x)}$$

However, as we take the limit of $dx$ the rate calculated becomes closer and closer to the slope at that point. For all practical purposes the $dx$ part is negligible and we can say that the slope is taken at $x$.

If you are not familiar with the formula above, I suggest you learn about limits first before doing calculus.

edited Nov 12 '19 at 16:34

Andrew Chin

7,389

answered Nov 12 '19 at 15:59

Sam

2,447

Then why do we say that derivative is a tangent as if it was a tangent then it couldn't be change between two points – Ritanshu Singh Nov 12 '19 at 16:02
Slope at a point and tangent are the same thing. – Sam Nov 12 '19 at 16:03

score 0 · Answer 3 · answered Nov 12 '19 at 16:06

When we talk about the rate of change of a function at a point, that's just shorthand for saying the rate of change near the point, or as the function passes through the point, as it were.

There are many such shorthand phrases in analysis, and you will need getting used to such informal turns of phrase. Another example that comes to mind is talking of the limit of a function at a point, or at infinity. Of course what we mean is actually the limit of the function as we approach that point, or near that point.

These are some of the things you need to get used to, especially when the language is not formal.

How can we define the rate of change of a function at a single point?

3 Answers3