So in the following LinkedIn post I got some idiotic comments from both Arthur Rubin and Charles Roddie who were struggling to understand my historic geometric theorem.

They didn't bother to study my article, but just like the silly monkeys they are, they started plugging in numbers without understanding what they are doing.

In the end, I showed them how to do the examples:

https://www.linkedin.com/feed/update/urn:li:activity:6876769312925900800

In one comment I did the complete example:

I will use f(x)=x^2 and f'(x)=2x.

In which case Q(x,h)=h.

{ For f(x)=x^4, Q(x,h) = 6x^2h + 4xh^2 + h^3 and I will leave this as an exercise for you!}

f(x)=x^2 f'(x)=2x Let x=1, h=n=2

f(x+h)-f(x) = 2Q(1,2) + 2f'(1)
f(3)-f(1) = 2(2) + 2(2)

8 = 8

Q(x,1)+Q(x+1,1)=f(x+2)-f(x)-f'(1)-f'(2)
-> f(x+2)-f(x) = Q(x,1)+Q(x+1,1)+ f'(1)+f'(2)
f(3)-f(1) = 1 + 1 + 2 + 4

8 = 8

As expected.

Late Edit: I decided to do your example for you:

f(x)=x^4
f'(x) = 4x^3 with h=n=3 and x=0

Q(x,h) = 6x^2h + 4xh^2 + h^3

f(0+3)-f(0) = { f'(0)+f'(1)+f'(2)} + { Q(0,1)+Q(1,1)+Q(2,1) }

81 - 0 = (0 + 4 + 32) + (1+11+33)

81 = 36 + 45

Again, exactly as expected.

-----------------------------------------------------------------------------------------------------------

The thing about theorems is that they are TRUE. It seems that Arthur Rubin and Charles Roddie still have a long way to go....

A general derivative formula for all smooth functions:

https://lnkd.in/gXwsXBTR

OR

https://lnkd.in/evDiUHWz

Fixing the broken mainstream calculus:

https://lnkd.in/gb_aemPs

OR

https://lnkd.in/e9RmYqtr