# Differential and square of differential

by rsaad
Tags: differential, square
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 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? ($\frac{dy}{dx}$) 2 = (d2 y)/(dx2)
 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.
 P: 295 The equality that does hold is $\displaystyle \left( \frac{d}{dx} \right)^2y = \frac{d^2y}{(dx)^2}$, which is usually abbreviated (somewhat abuse of notation) as $\displaystyle \frac{d^2y}{dx^2}$.
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P: 39,503
Differential and square of differential

 Quote by rsaad 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? ($\frac{dy}{dx}$) 2 = (d2 y)/(dx2)
I can't help but wonder where you "often see" that? As others said, it is certainly NOT "generally" true. $d^2y/dx^2= (dy/dx)^2$ is a "differential equation". If we let u= dy/dx, we have the "first order differential equation" $du/dx= u^2$ which can be written as $u^{-2}du= dx$ and, integrating, $-u^{-1}= x+ C$ so that $u= dy/dx= -\frac{1}{x+ C}$. Integrating that, $y(x)= \frac{1}{(x+ C)^2}+ C_1$. Only such functions satisfy your equation.
P: 759
 Quote by HallsofIvy $u= dy/dx= -\frac{1}{x+ C}$. Integrating that, $y(x)= \frac{1}{(x+ C)^2}+ C_1$. Only such functions satisfy your equation.
Integrating -1/(x+C) leads to -ln(x+C)+c = -ln(ax+b)
P: 295
 Quote by JJacquelin 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.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,503 Yeah, I confuse very easily! Thanks.
P: 234
 Quote by HallsofIvy Yeah, I confuse very easily! Thanks.
You do get sloppy very often, Halls. Shape up!

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