Second derivative notation

[ImAnEngineer]

I often see the second derivative written down like this:

$$\frac{d^2y}{dx^2}$$

Although it seems more logical to me to write

$$\frac{d^2y}{d^2x^2}$$

Or

$$\frac{d^2y}{(dx)^2}$$

Since it represents

$$\frac{d}{dx} \frac{dy}{dx}$$

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

 

[tiny-tim]

Hi ImAnEngineer!

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.

 

[workmad3]

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 :)

 

[ImAnEngineer]

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

Is it?

I would say:

$$\left(\frac{d}{dx}\right)^n=\frac{d^n}{(dx)^n}=\frac{d^n}{d^nx^n}$$

Because (ab)²=a²b² and not ab²

So is it just a shortcut notation to leave out the ² in the denominator?

 

[slider142]

Is it?

I would say:

$$\left(\frac{d}{dx}\right)^n=\frac{d^n}{(dx)^n}=\frac{d^n}{d^nx^n}$$

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).

 

[yenchin]

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...

 

[tiny-tim]

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!

 

[ImAnEngineer]

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!