Differentiatial equation

[TimeRip496]

Does $$ ∅*(\frac{d}{dξ})=∅*(\frac{d1}{dξ}) $$?

If is true,
Does multiplying a function and a derivative equals to the derivative of that function? For e.g. $$ ∅*(\frac{d}{dξ})=\frac{d∅}{dξ} $$ where ∅ is a function of ξ

But isn't it supposed to be like this(based on the product rule), $$ ∅*(\frac{d}{dξ}) = ∅*(\frac{d1}{dξ}) = \frac{d}{dξ}*∅-1*\frac{d∅}{dξ} $$ ?

What if ∅ is a constant or is not a function of ξ?

 

[blue_leaf77]

Does $$ ∅*(\frac{d}{dξ})=∅*(\frac{d1}{dξ}) $$?

Obviously no. Derivative symbol with nothing next to the right of it constitutes no meaningful quantities, no numerical value can be associated with it (if the variable is given a number), it's just an instruction to differentiate whatever stands on the right. If you put something to the right of a derivative (like you did in the RHS of that equation), you have given a numerical value to the entire expression.
Therefore
$$
∅*(\frac{d}{dξ})\neq \frac{d∅}{dξ}
$$

 

[TimeRip496]

Obviously no. Derivative symbol with nothing next to the right of it constitutes no meaningful quantities, no numerical value can be associated with it (if the variable is given a number), it's just an instruction to differentiate whatever stands on the right. If you put something to the right of a derivative (like you did in the RHS of that equation), you have given a numerical value to the entire expression.
Therefore
$$
∅*(\frac{d}{dξ})\neq \frac{d∅}{dξ}
$$

Thanks!

But then how do I get from equation (12) to equation (13)? The only way I can do it is when
$$
∅*(\frac{d}{dξ}) = \frac{d∅}{dξ}.
$$

 

[blue_leaf77]

Where did you get source from? Is it the same source as the one with harmonic oscillator in another thread of yours?

 

[TimeRip496]

Where did you get source from? Is it the same source as the one with harmonic oscillator in another thread of yours?

Yes.
Source: http://vixra.org/pdf/1307.0007v1.pdf

 

[blue_leaf77]

I believe that's not the common and standard way to write the derivative of a function; in equation (12), ##\phi_0## should be on the right of the bracketed terms.

 

[TimeRip496]

I believe that's not the common and standard way to write the derivative of a function; in equation (12), ##\phi_0## should be on the right of the bracketed terms.

Ok thanks again for your help!