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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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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

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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Could you please cite a specific book that says it is a fraction?Some books will mention that dy/dx is a symbol some say its a fraction,wht the truth? please help

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

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

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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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A truly exotic configuration of symbols there, HallsofIvy...[tex]\frac{dy}{dx}= \lim_{y\to 0}\frac{y(x+y)- y(x)}{h}[/tex]

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HallsofIvy

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Yes, that particular definition of derivative is only known to us really advanced people!

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