Some books will mention that dy/dx is a symbol

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Some books will mention that dy/dx is a symbol some say its a fraction,wht the truth? please help
 

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Mentallic
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It is NOT a fraction. Some properties of derivatives behave in the same way as fractions (which is why many people tend to drift away from its true meaning and more towards it simply being a fraction) but you need to recognize and understand the proofs behind these properties to realize that it's not simply "because that's how fractions work".

[tex]\frac{dy}{dx}=\lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x}=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]

While you too might drift away from thinking of it as a limit as treat it more like a fraction, while doing so, just appreciate that you can do so :wink:


p.s. There are other symbols to represent the derivative, such as y', f'(x) etc. So I'm guessing that the expression dy/dx was coined because it does make things simpler when using the chain rule and such.

[tex]\frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx}[/tex] is much easier to remember than [tex]f'(x)=g'(h(x))h'(x)[/tex] where [tex]y=g(u)[/tex] and [tex]u=h(x)[/tex]

Even though solving derivatives will be quickly done in your head this way, the representation of derivatives is much simpler to follow - especially when they get harder - the other way.
 
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HallsofIvy
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Some books will mention that dy/dx is a symbol some say its a fraction,wht the truth? please help
Could you please cite a specific book that says it is a fraction?

You should be aware of a distinction between "derivatives" and "differentials".
[tex]\frac{dy}{dx}= \lim_{h\to 0}\frac{y(x+h)- y(x)}{h}[/tex]
, the differential, is NOT a fraction but, since it is the limit of a fraction, we can often go back "before" the limit, use the fraction property, the come "forward", taking the limit to show that it can be treated like a fraction.

To make use of that, we can define "dx" as a symbol, define dy= y'(x)dx and then define the "fraction" dy/dx. It is still not a "true" fraction because the "numerator" and "denominator" are not numbers or functions. I use y'(x) rather than dy/dx above so as not to confuse the two different uses of "dy/dx".
 
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Try not to look at it as a fraction (like everyone has already stated), but look at it as a ratio of change between two variables. Usually, x & y....or otherwise stated, how x changes in relation to y...a ratio of change.
 
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arildno
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[tex]\frac{dy}{dx}= \lim_{y\to 0}\frac{y(x+y)- y(x)}{h}[/tex]
A truly exotic configuration of symbols there, HallsofIvy...:devil:
 
  • #6
HallsofIvy
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Yes, that particular definition of derivative is only known to us really advanced people!
 

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