Differential and square of differential

In summary, the equality (dy/dx)^2 = d^2y/dx^2 does not hold in general, and is only true for the specific family of functions y = -ln(ax+b). The correct equality is (d/dx)^2y = d^2y/dx^2.
  • #1
rsaad
77
0
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 = (d2 y)/(dx2)
 
Physics news on Phys.org
  • #2
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
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
rsaad said:
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 = (d2 y)/(dx2)
I can't help but wonder where you "often see" that? As others said, it is certainly NOT "generally" true. [itex]d^2y/dx^2= (dy/dx)^2[/itex] is a "differential equation". If we let u= dy/dx, we have the "first order differential equation" [itex]du/dx= u^2[/itex] which can be written as [itex]u^{-2}du= dx[/itex] and, integrating, [itex]-u^{-1}= x+ C[/itex] so that [itex]u= dy/dx= -\frac{1}{x+ C}[/itex]. Integrating that, [itex]y(x)= \frac{1}{(x+ C)^2}+ C_1[/itex]. Only such functions satisfy your equation.
 
  • #5
HallsofIvy said:
[itex]u= dy/dx= -\frac{1}{x+ C}[/itex]. Integrating that, [itex]y(x)= \frac{1}{(x+ C)^2}+ C_1[/itex]. Only such functions satisfy your equation.
Integrating -1/(x+C) leads to -ln(x+C)+c = -ln(ax+b)
 
  • #6
JJacquelin said:
Integrating -1/(x+C) leads to -ln(x+C)+c = -ln(ax+b)

He probably got confused and differentiated instead. Anyways, the identity given in the OP is not generally true, actually, it is almost surely false, given the family of functions that can be defined on R2.
 
  • #7
Yeah, I confuse very easily! Thanks.
 
  • #8
HallsofIvy said:
Yeah, I confuse very easily! Thanks.
You do get sloppy very often, Halls. Shape up! :smile:
 

What is a differential?

A differential is a mathematical concept that represents the instantaneous rate of change of a variable with respect to another variable. It is typically denoted by the symbol "d" and can be thought of as an infinitely small change in a variable.

What is the square of a differential?

The square of a differential is the product of a differential with itself. It is often used in calculus to represent a second derivative or a rate of change of a rate of change.

How is a differential different from a derivative?

A differential is a notation used to represent a derivative, whereas a derivative is a mathematical operation that calculates the instantaneous rate of change of a function. In other words, a differential is a way of writing a derivative, but they are not the same thing.

What is the chain rule for differentials?

The chain rule for differentials is a rule in calculus that allows us to calculate the derivative of a composition of functions. It states that the differential of a composite function is equal to the product of the differentials of each individual function multiplied together.

How are differentials and integrals related?

Differentials and integrals are inverse operations in calculus. The integral of a function represents the summation of all the infinitely small changes in that function, while a differential represents an infinitely small change in a function. In other words, we can use integrals to find the total change in a function, and differentials to find an infinitely small change in a function.

Similar threads

  • Calculus
Replies
2
Views
1K
Replies
46
Views
804
Replies
14
Views
1K
Replies
22
Views
2K
Replies
1
Views
896
Replies
15
Views
1K
Replies
4
Views
902
Replies
2
Views
956
Replies
6
Views
2K
Replies
10
Views
900
Back
Top