User:ExpiredVenom

Welcome to my userpage!
I am ExpiredVenom (if you couldn't tell already).

This does not use standard mathematical notation (to avoid unwieldiness)
$$^1_x\Delta_nf(x)\equiv f(x+n)-f(x)$$

$$^c_x\Delta_nf(x)\equiv^1_x\Delta_n(^{c-1}_x\Delta_nf(x))$$

Second derivative of a quadratic function
$$f(x)=ax^2+bx+c$$ $$\dfrac{d}{dx}f(x)=\dfrac{d}{dx}(ax^2+bx+c)$$ $$\dfrac{d}{dx}f(x)=2ax+b$$ $$\dfrac{d^2}{dx^2}f(x)=\dfrac{d}{dx}(2ax+b)$$ $$\dfrac{d^2}{dx^2}f(x)=2a$$ Now we have established the second derivative of a quadratic function to be $$2a$$.

Second difference of a quadratic function
$$_x\Delta_1f(x)=_x\Delta_1(ax^2+bx+c)=2ax+a+b$$

$$^2_x\Delta_1f(x)=_x\Delta_1(_x\Delta_1(ax^2+bx+c))=_x\Delta_1(2ax+a+b)$$

$$_x\Delta_1(2ax+a+b)=2a$$