Second derivative notation

  • #1
209
1

Main Question or Discussion Point

I often see the second derivative written down like this:

[tex]\frac{d^2y}{dx^2}[/tex]

Although it seems more logical to me to write

[tex]\frac{d^2y}{d^2x^2}[/tex]

Or

[tex]\frac{d^2y}{(dx)^2}[/tex]

Since it represents

[tex]\frac{d}{dx} \frac{dy}{dx} [/tex]

Is there any logic behind this or is it just a shortcut notation to omit the square in d², or brackets in the denominator?
 

Answers and Replies

  • #2
tiny-tim
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Hi ImAnEngineer! :smile:

It's because it's short for (d/dx)2(y) …

for example, you might write (d/dx)2(x3 + sinx), or indeed (d/dx)28(x3 + sinx) …

and (d/dx)n is naturally written without brackets as dn/dxn

the x3 + sinx stays as it is. :wink:
 
  • #3
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I think that if the notation had d2x2 then people may be tempted to do silly things like cancel the d2 and the x2 and get really confused :) As it is, there's only a slight bit of confusion in areas such as this :)
 
  • #4
209
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...
and (d/dx)n is naturally written without brackets as dn/dxn
Is it?

I would say:

[tex]\left(\frac{d}{dx}\right)^n=\frac{d^n}{(dx)^n}=\frac{d^n}{d^nx^n}[/tex]

Because (ab)²=a²b² and not ab²

So is it just a shortcut notation to leave out the ² in the denominator?
 
  • #5
1,013
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Is it?

I would say:

[tex]\left(\frac{d}{dx}\right)^n=\frac{d^n}{(dx)^n}=\frac{d^n}{d^nx^n}[/tex]

Because (ab)²=a²b² and not ab²

So is it just a shortcut notation to leave out the ² in the denominator?
Semantically, d2x2 may imply that the differential operator is being applied to x twice, which is not the case in (dx)2. Ie., it is like mistaking (sin x)2 for sin2x2.
In the case of writing dx2, it is just treating dx as a single entity, not as d(x2).
 
  • #6
540
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It is the same in differential geometry and relativity, where line element (metric) is written as ds^2 instead of (ds)^2. It save some works in writing I suppose...
 
  • #7
tiny-tim
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In the case of writing dx2, it is just treating dx as a single entity, not as d(x2).
It is the same in differential geometry and relativity, where line element (metric) is written as ds^2 instead of (ds)^2. It save some works in writing I suppose...
Yup! :biggrin:
 
  • #8
209
1
Semantically, d2x2 may imply that the differential operator is being applied to x twice, which is not the case in (dx)2. Ie., it is like mistaking (sin x)2 for sin2x2.
In the case of writing dx2, it is just treating dx as a single entity, not as d(x2).
Aah OK! This makes sense, that really helps.

Thanks everyone! :smile:
 

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