Higher Level Derivative Notation

[SpaceRocks]

Hi,

With respect to derivative notation...

d/dx(y) //1st derivative of y w.r.t x
d/dx (dy/dx) = d^2y/dx^2 //2nd derivative
d/x (d^2y/dx^2) = d^3y/dx^3 //3rd derivative

If you continue finding derivatives in this way, why do the d's increment in the numerator and not in the denominator while the x's increment in the denominator?

I understand the pattern and even intuitively it makes sense to read: d^2y/dx^2 as the second derivative of y w.r.t x, but I don't understand why the notation behaves this way.

*The formatting changed when I posted. The "//" means comment.

Thanks!

 

[Buzz Bloom]

Hi SpaceRocks:

It took a bit of searching but I found it. Leibniz invented this notation in the 17th century. See

https://en.wikipedia.org/wiki/Leibniz's_notation​

section "Leibniz notation for higher derivatives".

Regards,
Buzz

 

[Ssnow]

Hi SpaceRocks, your question is interesting because this choice of notation always create confusion in my mind. There was a period where I belived that $\frac{d^2}{dx^2}y(x)$ was $\frac{d^2}{(dx)^2}y(x)$ but the notation is not clear ... I think that one of the best way to denote the derivative, at list for the first approach, is $D_{x}(\cdot)$, so the second derivative is $D_{x}(D_{x}(\cdot))=D_{x}^2(\cdot)$ and so on ...

Ssnow

 

[SpaceRocks]

It looks like it's just a customary notation and it behaves the way it does because that's how it's structured. From the Wiki article, the Use of Various Forms section makes sense of it for me.

Thanks a lot!