Question about Leibniz's notation for derivatives

[paul2211]

Hi, I'm a new member to the forum, and I'm currently studying Calculus.

Basically, derivatives can be written as (dy/dx) in Leibniz's notation, but I remember my teacher saying that it's just a symbol and shouldn't be used like two variables (dy and dx)...

However, when there's some integral and inside it, there's a (dy/dx) * (dx), the teacher says we can cancel the two dx, which contradicts what he said earlier.

Also, when we wish to find the integral of (dy/dx) = x, he said we can multiply by dx on both sides, which of course is also confusing me...

So can you guys help me out, and explain these things to me?

Thanks very much :D

 

[dextercioby]

The Leibniz notation is indeed confusing, makes you think you can fool around with the 'd'-s as if they were numbers. It's false, of course.

What you should remember is that

$$y(x)+C=\int \left(\frac{d}{dx}y(x)\right){}dx$$

So no canceling, no nothing, the notation with fractions is just what it is, a notation, not a ratio.

 

[lugita15]

Actually, the original name of calculus was the Infinitesimal Calculus. Calculus was thought to be the study of manipulating infinitely small quantities. So dx is an infinitely small amount of x, and dy is an infinitely small amount of y. These are known as differentials, and the derivative was thought to be a ratio of differentials. The original definition of the derivative was df/dx=(f(x+dx)-f(x))/dx. So for Leibniz the chain rule really was just multiplying fractions, as was the fundamental theorem of calculus.

Then later, people decided they didn't like the infinitesimal methods of Newton and Leibniz, so they invented more rigorous methods like limits (specifically the epsilon-delta definition).

Recently Abraham Robinson found a way to rigorously justify infinitesimal methods, but that's irrelevant. The more important point is that intuitive notions of infinitesimals will make calculus MUCH easier to make sense of (plus they'll help you in physics courses, where familiarity with infinitesimals is often taken for granted.). If you're interested in this approach, I'd recommend Calculus Made Easy by Silvanus Thompson, a short little book which is a century old but is still as relevant as ever. Or Calculus Without Limits by John C. Sparks, if you want a more conventional textbook.

 

[arildno]

That the derivatives in Leibniz notation cannot be naively understood as fractions is obvious from identities like:
$$\frac{\partial{x}}{\partial{y}}\frac{\partial{y}}{\partial{z}}\frac{\partial{z}}{\partial{x}}=-1$$

 

[lugita15]

That the derivatives in Leibniz notation cannot be naively understood as fractions is obvious from identities like:
$$\frac{\partial{x}}{\partial{y}}\frac{\partial{y}}{\partial{z}}\frac{\partial{z}}{\partial{x}}=-1$$

That's just because the notation we use for partial derivatives is a bit misleading, but we can still think of partial derivatives as ratios of infinitesimals. For instance, $$\frac{\partial f}{\partial x} = \frac{f(x+dx,y)-f(x,y)}{dx}$$. If we use definitions like this, the formula you gave makes perfect sense.

 

[ManishR]

if dy/dx = dy divided by dx which is 0/0.

as earlier posts said it is just a symbol its behave like it is "dy divided by dx". but its not.

 

[lugita15]

if dy/dx = dy divided by dx which is 0/0.

as earlier posts said it is just a symbol its behave like it is "dy divided by dx". but its not.

No, it represents an infinitely small quantity divided by an infinitely small quantity, which is NOT the same as 0/0.

 

[ManishR]

No, it represents an infinitely small quantity divided by an infinitely small quantity, which is NOT the same as 0/0.

$$\frac{df(x)}{dx}=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}$$

h is not infinitely small.

i always find it {}``non-mathematical'' and confusing.

infintely small = smallest difference = 0

h is a variable such as

$$h\in R-\{0\}$$

 

[lugita15]

$$\frac{df(x)}{dx}=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}$$

h is not infinitely small.

i always find it {}``non-mathematical'' and confusing.

infintely small = smallest difference = 0

h is a variable such as

$$h\in R-\{0\}$$

Look up nonstandard analysis, or just read Calculus Made Easy.

 

[JonDrew]

No, it represents an infinitely small quantity divided by an infinitely small quantity, which is NOT the same as 0/0.

Wait, so if dx is infinitely small and it is multiplied into a function wouldn't that make the function also infinitely small?

Leibniz notation is something that has been baffling me lately, I think I am going to have to read the Silvanus Thompson book you suggested.