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Differential and square of differential 
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#1
Nov2212, 08:54 AM

P: 77

Hi
I often see the following in books but I do not understand how they are equal. So can someone please tell me for what conditions does the following equality hold? ([itex]\frac{dy}{dx}[/itex]) ^{2} = (d^{2} y)/(dx^{2}) 


#2
Nov2212, 09:11 AM

P: 759

No, generaly this equality doesn't hold.
For example : y=x² dy/dx = 2x d²y/dx² = 2 (dy/dx)² = (2x)² = 4x² 2 is not equal to 4x² The equality (dy/dx)² = d²y/dx² holds only for one family of functions : y =  ln(ax+b). It doesn't hold for any other function. 


#3
Nov2212, 09:28 AM

P: 295

The equality that does hold is [itex]\displaystyle \left( \frac{d}{dx} \right)^2y = \frac{d^2y}{(dx)^2}[/itex], which is usually abbreviated (somewhat abuse of notation) as [itex]\displaystyle \frac{d^2y}{dx^2}[/itex].



#4
Nov2212, 11:20 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,497

Differential and square of differential



#5
Nov2212, 12:42 PM

P: 759




#6
Nov2212, 03:07 PM

P: 295




#7
Nov2312, 08:13 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,497

Yeah, I confuse very easily! Thanks.



#8
Nov2312, 01:00 PM

P: 234




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