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How to substitute a function in a derivative

asked 5 years ago

Cyrille gravatar image

I know that if U(x,y)=Axαyβ, I have Ux=αAxα1yβ. But by substitution, I can obtain Ux=α(Ux). Is there a way to obtain this in Sagemath. I have tried

U_x= U.diff(x).subs(Ax^(alpha)y^(beta)== U)

but this has no effect.

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answered 5 years ago

Emmanuel Charpentier gravatar image

If I understand you correctly, you meant that Ux=Ux, right ? If so, your result is a triviality, since :

  • There exists some quantity K such as U(x,y)=Kxα, where K doesn't depend on x (in fact, it is obvious that K=Ayβ).

  • Therefore Ux=Kxαx=Kαxα1

This can be checked in sage quite directly:

sage: U(x,y)=A*x^a*y^b;U
(x, y) |--> A*x^a*y^b
sage: bool(U(x,y).diff(x)==a*U(x,y)/x)
True
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Of course Emmanuel It's trivial. But for students it is not. You propose a verification not a substitution neither a demonstration.

Cyrille gravatar imageCyrille ( 5 years ago )

Ah. I see what you mean. Bt that's not substitution per se. What you're trying to do is:

sage: var("z")
z
sage: solve(U.diff(x)(x,y)==z*U(x,y),z)
[z == alpha/x]

I can't say I find trhis especially illuminating...

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 5 years ago )

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Asked: 5 years ago

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Last updated: Sep 21 '19