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Second Derivative Theorems

  1. Jul 8, 2005 #1


    I am posting my theorems for peer review, anyone interested in posting some proofs using some simple functions?

    Can these theorems be reduced into simpler equations?

    Orion1 Second Derivative Theorems:
    [tex]\frac{d^2}{dx^2} (x) = 0[/tex]
    [tex]\frac{d^2}{dx^2} (x^2) = 2[/tex]
    [tex]\frac{d^n}{dx^n} (x^n) = n![/tex]
    [tex]\frac{d^2}{dx^2} (x^n) = n(n - 1) x^{n - 2}[/tex]
    [tex]\frac{d^2}{dx^2} (x^{-n}) = n(n + 1)x^{-n - 2}[/tex]

    [tex]\frac{d^2}{dx^2} \left[ f(x) \pm g(x) \right] = \frac{d^2}{dx^2} [f(x)] \pm \frac{d^2}{dx^2} [g(x)][/tex]

    [tex]\frac{d^2}{dx^2} [f(x) \cdot g(x)] = \frac{d^2}{dx^2} [f(x)] \cdot g(x) + 2 \frac{d}{dx} [f(x)] \cdot \frac{d}{dx} [g(x)] + \frac{d^2}{dx^2} [g(x)] \cdot f(x)[/tex]

    [tex]\frac{d^2}{dx^2} \left[ \frac{f(x)}{g(x)} \right] = \frac{\frac{d^2}{dx^2} [f(x)] \cdot [g(x)]^2 - 2 \frac{d}{dx} [f(x)] \cdot g(x) \cdot \frac{d}{dx} [g(x)] + \left[ g(x) \cdot \frac{d^2}{dx^2} [g(x)] - 2 \left( \frac{d}{dx} [g(x)] \right)^2 \right] \cdot f(x)}{[g(x)]^3}[/tex]

     
  2. jcsd
  3. Jul 8, 2005 #2
    [tex]\frac{d^n}{dx^n} (x^n) = n![/tex]

    Ah, at first I disagreed. But now I see it. I like that one.
     
    Last edited: Jul 8, 2005
  4. Jul 8, 2005 #3

    robphy

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    The pattern in [tex]\frac{d^2}{dx^2} [f(x) \cdot g(x)] = \frac{d^2}{dx^2} [f(x)] \cdot g(x) + 2 \frac{d}{dx} [f(x)] \cdot \frac{d}{dx} [g(x)] + \frac{d^2}{dx^2} [g(x)] \cdot f(x)[/tex]
    is more easily seen using the "prime" notation:
    [tex] (fg)'' = f''g+2f'g'+fg''[/tex]
    ...the coefficients are just like those in
    [tex] \begin{align*}
    (f+g)^2
    &= f^2g^0+2f^1g^1+f^0g^2
    \end{align*}[/tex]
     
  5. Jul 9, 2005 #4

    lurflurf

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    [tex]\frac{d^2}{dx^2}u^v=2u^{v-1}\frac{du}{dx}\frac{dv}{dx}+v(v-1)u^{v-2}(\frac{du}{dx})^2+v u^{v-1}\frac{d^2u}{dx^2}+u^v\log^2(u)(\frac{dv}{dx})^2+u^v\log(u)\frac{d^2v}{dx^2}[/tex]
     
  6. Jul 30, 2005 #5
    functional malfunction...

    lurflurf theorem:
    [tex]\frac{d^2}{dx^2}u^v=2u^{v-1}\frac{du}{dx}\frac{dv}{dx}+v(v-1)u^{v-2}\left(\frac{du}{dx}\right)^2+vu^{v-1}\frac{d^2u}{dx^2}+u^v\log^2(u)\left(\frac{dv}{dx}\right)^2+ u^v\log(u)\frac{d^2v}{dx^2}[/tex]



    lurflurf, your theorem appears to be missing a factor: [tex][1 + v \ln(u)][/tex]

    Orion1 second derivative theorem:
    [tex]\frac{d^2}{dx^2}u^v=2u^{v-1}[1+v\ln(u)]\frac{du}{dx}\frac{dv}{dx}+v(v-1)u^{v-2}\left(\frac{du}{dx}\right)^2+vu^{v-1}\frac{d^2u}{dx^2}+u^v\ln^2(u)\left(\frac{dv}{dx}\right)^2+u^v\ln(u)\frac{d^2v}{dx^2}[/tex]

    Is this theorem correct?

     
    Last edited: Jul 30, 2005
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