- #1

Orion1

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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]