What exactly is the idea behind the notion, "dy/dx"?

I am a Calculus noob so the best answer will be chosen for the easy explanations.



I am very familiar with

f'(x) = lim(h-%26gt;0) f(x+h)-f(x)/h. In fact, I was able to understand most of diffrenciation stuff with this rule. I was able to understand quotient rule

and the the product rule using f'(x) idea.



but when it comes to the notation dy/dx, I am very confused.

The statement, %26quot;change in y in respect to the x%26quot;....I really don't know what it is saying. so I have trouble understanding how the chain rule works.



dy/dx = dy/du * du/dx... The textbook says If u changes twice as fast as x and y changes three times as fast as u, then it seems reasonable that y changes six times as fast as x..... I mean wtf is it saying? You have to know this idea to know how to do related rates and implicit.



I mean du's canceling each other to make the sentence work so that it gets dy/dx.... uh ???????? whaT??? (in the chain rule formulae)



Calculus would be so much easier if the idea dy/dx is gone.

oh man, I am going to have so much trouble with the related rates topics.



help me to get this idea, dy/dx!!What exactly is the idea behind the notion, %26quot;dy/dx%26quot;?
dy is extremely small change in y.

dx is extremely small change in x.

dy/dx = lim delta x tends to 0 ( delta y / delta x ).

suppose, dy/dx = 3x

then, dy = 3x dx, which answers your chain rule question,

i hope it helps you.What exactly is the idea behind the notion, %26quot;dy/dx%26quot;?
You've got several problems on top of each other here. I can't look at all of them but here's a start:



Think of the set of squiggles (dy/dx) as just the name of an expression such as (2x-3) or (sin(x) / 4) etc. Just a name. Nothing else. It's a placeholder for a function that you can move around an equation but it is NOT the eexpression itself.



Now, personally, when I name things I like to call them things like Fido or Matilda but mathematicians are a strange lot. It was probably a mathematician who named things in the Starwars movies with names like %26quot;r2d2%26quot;.



Anyway, Fido (dy/dx to you) is the name of the derivative expression of whatever you are dealing with. Eg.

Fido = [d(x^2) / dx] means that Fido = (2x).



The expression that Fido stands for is a normal, counting numbers, real-world thing. Not an infinitely small something.



A real-world thing only becomes infinitely small when you multiply it by dx or dy or d%26lt;something else%26gt;.



When we want to integrate something, we want to add up a gazillion infinitely small bits of it and that's why an integrand always ends with %26quot;dx%26quot; for example. You have a real-world expression followed by %26quot;.dx%26quot; to change it into an infinitely small thing we can integrate.



And this is where I have to admit there's method in the (dy/dx) madness because if instead of calling (2x) %26quot;Fido%26quot;, you call it %26quot;dy/dx%26quot; - which looks like a fraction - then you can start doing fraction'ish things with it in equations.



For example, you can multiply both sides of an equation by the bottom bit %26quot;dx%26quot;. Eg:



dy/dx = 2x (We can't integrate (2x) and we can't integrate (dy/dx))

dy = 2x.dx (but we can now integrate (2x.dx) and we can integrate dy as well.



Loking at chain rule stuff [dy/du]*[du/dx] is just a way of saying %26quot;I want to multiply these two expressions%26quot; - but it (obviously) doesn't actually tell you what those expressions are. To get the real dirt, you have to replace the placeholders %26quot;dy/du%26quot; and %26quot;du/dx%26quot; with the actual expressions they stand for.What exactly is the idea behind the notion, %26quot;dy/dx%26quot;?
dy/dx is geometrically a tangent to the curve(function).What exactly is the idea behind the notion, %26quot;dy/dx%26quot;?
dy/dx is the same as f'(x). you may be comfortable with using that notation. when you write dy/dx, you are differentiating y with respect to x. note that y=x^2, for example, is expressing y as a function of x. the same equation could be written as f(x)=x^2. Differentiating would give f'(x)=2x, or dy/dx=2x. The term %26quot;dx%26quot; is exactly that, just a term, known as the differential. By themselves, however, dy and dx don't make much sense. It is their ratio, dy/dx that expresses the slope of the line tangent to the curve at a given point, or as a limit, as you already noted you were familiar with.What exactly is the idea behind the notion, %26quot;dy/dx%26quot;?
dy/dx is actually notation for calculus created by Gottfried Leibniz.

About the time Newton was ready to publish his work on calculus Leibniz published his work on the same subject.

this actually caused a huge controversy in europe at the time and even though to this day we generally accredit newton with the discovery- they both 'invented' calculus independent from each other.

When comparing the works of each, they each had different notations, Newton used what he called 'fluxions' which look like the Greek symbol theta (sort of) with the prime mark to show changes, shifts in movement instantaneously (this symbol ' is prime).

Leibniz expressed calculus as the 'study of infinitesimals' and in fact used d/dx and dy/dx to represent the %26quot;zooming%26quot; effect (he actually had a whole bunch of notations; proabbly why we nowadays have so many ways of writing derivatives) which zeroes in at a certain point in motion, half-life, area et cetera. Leibniz was more careful and his notation is actually more to the point [but again, we accredit Newton moreso].

Similarly, l'Hopital's rule in calc wasn't invented by LHopital but published in large scale by his firm- even though the method had been used for decades- he is accredited w the governing law in calculus.





there is a lot more on this topic if you are interested..... check the link and whatnotWhat exactly is the idea behind the notion, %26quot;dy/dx%26quot;?
dy means an infinitesimal (very very small) change in y.



dy/dx is the rate of change of y per infinitesimal change in x. It's easy to see in a straight line slope.



So if a line has a slope of 3, then for every change of 1 in u, y changes by 3. So dy/du = 3. But imagine that for other curves on a really small scale.



If u increases by 2 for every change in x by 1, then du/dx is 2. You can tell then for every change of x by 1, u changes by 2, for each of those, y changes by 3. So the total change in y is 6 per change in x. Thus dy/dx is 6.What exactly is the idea behind the notion, %26quot;dy/dx%26quot;?
Chill out !



I feel that you are reading too much into the theory.



The exercises you do will not delve deeply into the topics you have raised.



Relax !What exactly is the idea behind the notion, %26quot;dy/dx%26quot;?
I'll bet you understand the concept of deltay/delta x.

dy/dx is simply the limit (if it exists) of delta y/delta x as delta x approaches zero. You can think of dy/dx as a fraction as long as dx not = 0.What exactly is the idea behind the notion, %26quot;dy/dx%26quot;?
Personally I really like the dy/dx notation because it makes so much sense!



You see when you calculate derivitives you are calculating the gradient:



(f(x+h) - f(x)) / h is just %26quot;rise over run%26quot;, how much the slope changes.



dy is the amount that y changes

dx is the amount that x changes



so

dy/dx says %26quot;change in y divided by the change in x%26quot; or more simply %26quot;rise over run%26quot;, how much up and how much across.





%26quot;dy/dx = dy/du * du/dx... The textbook says If u changes twice as fast as x and y changes three times as fast as u, then it seems reasonable that y changes six times as fast as x..... I mean wtf is it saying?%26quot;



dy/du is how much y changes over how much u changes, so if y goes up 3 for every 1 u goes across then dy/du = 3

du/dx is how much u changes over how much x changes, so if u goes up 2 for every 1 x goes across then du/dx = 2

The chain rule tells us that as a result dy/dx = 6 or %26quot;y goes up 6 for every 1 x goes across



You have to know this idea to know how to do related rates and implicit.

Yes the chain rule is key to understanding implicit differentiation and related rates.



Good luck, if you need help we are just a few mouse clicks away.