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When I was studying basic differential calculus I found out that when $y=f(x)$ the $dy$ is the change in $y$ when $x$ is changed to $x+dx$.

Now when I was studying physics it seems we add or multiply these infinitesimals like numbers and we integrate them, even in chaim rule

$$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$$

now here the infinitesimals are considered as numbers and they are freely mulitplied anywhere. For example if we are given an equation $y(x)=x^2$ we take the derivate both sides and find the corresponding change in $y$ when $x$ changes by $dx$.

Multiplying these terms makes absolutely no sense to me. Lets say we have and equation $y=x$ and we multiply both sides by $dx$ what does this represent where has the change been taken and the equation does not make any intuitive sense to me.

I have seen a very similar question but as I am a student of class 8th they don't make a lot of sense to me.

All answers are welcome, Thanks

amWhy
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Naruto
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  • Have you heard of 3blue1brown? He has a video series on calculus that explains these dy/dx infinitesimals very well: https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr – zhuli Jun 13 '20 at 09:35
  • Thanks man zhuli – Naruto Jun 13 '20 at 09:37
  • @Naruto See if this post helps you too. – Mrcrg Jun 13 '20 at 09:47
  • Well infinitesimals are not typical numbers which are studied in high school like rational or real or complex numbers. Most of the confusion in calculus comes by thinking these as real numbers. You should read about them in Keisler's book. – Paramanand Singh Jun 13 '20 at 15:40

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