# Second derivative notation

## Main Question or Discussion Point

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
Homework Helper
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. 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 :)

...
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?

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).

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
Homework Helper
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! 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! 