Second Derivative Theorems

[Orion1]

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:
$$\frac{d^2}{dx^2} (x) = 0$$
$$\frac{d^2}{dx^2} (x^2) = 2$$
$$\frac{d^n}{dx^n} (x^n) = n!$$
$$\frac{d^2}{dx^2} (x^n) = n(n - 1) x^{n - 2}$$
$$\frac{d^2}{dx^2} (x^{-n}) = n(n + 1)x^{-n - 2}$$

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

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

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

 

[Jameson]

$$\frac{d^n}{dx^n} (x^n) = n!$$

Ah, at first I disagreed. But now I see it. I like that one.

 

[robphy]

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

 

[lurflurf]

$$\frac{d^2}{dx^2}u^v=2u^{v-1}\frac{du}{dx}\frac{dv}{dx}+v(v-1)u^{v-2}(\frac{du}{dx})^2+v u^{v-1}\frac{d^2u}{dx^2}+u^v\log^2(u)(\frac{dv}{dx})^2+u^v\log(u)\frac{d^2v}{dx^2}$$

 

[Orion1]

functional malfunction...

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



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

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

Is this theorem correct?