Leibniz notation makes no sense?

[Pithikos]

I know this question has been out there many times. I read many threads already but I just didn't find a satisfactory answer. What some people say is that Leibniz notation is just a notation and not a fraction. Then we treat this notation as a fraction. But what's the reason to do it if a human can't understand it?

I study at the moment for my test and there is implicit differentiation, u-substitution etc and I can't learn them because I am stuck with this filthy notation.

say I have s(t)=10t²

its derivative is \frac{ds}{dt}=20t

I can treat that as a fraction and get ds=20t*dt

But what the heck does that mean?? The change of s is 20t times the change of t?

 

[tiny-tim]

Hi Pithikos!

I can treat that as a fraction and get ds=20t*dt

But what the heck does that mean?? The change of s is 20t times the change of t?

Yes!

What is worrying you about that? ​

 

[HallsofIvy]

Who told you a human can't understand it?

The first derivative of a function is NOT a fraction but it is a useful notation to write it like one because it can be treated like a fraction.

Specifically, the derivative is the limit of a fraction. To prove that the derivative has some specific property of a fraction, go back before the limit, use the property for the difference quotient, then take the limit again.

 

[mathwonk]

These answers are correct, but if it makes you feel any better, there is also an interpretation, to be given later in differential geometry, by which the fraction also makes sense as an actual fraction of things a little more abstract than numbers, and which can be understood by humans.

 

[arildno]

In his work on fluxions (i.e, derivatives), Newton warned about regarding them as proper fractions, and used the following analogy.

Suppose we look at the following proper fraction 2x/x=2, for some number x.
If we let x "grow" into infinity, the quantity will still equal 2, but we cannot any longer say that it is written in terms of a fraction between two NUMBERS (infinity not being a number)
Similarly the other way, by letting x shrink to 0, we no longer have any fraction to speak of, only the limit of such fractions.