This is a standard proof of the product rule:
Q.E.D.
I've been thinking about proving the product rule, and I'm not sure if I can use limits in this way to do it:
A)
or should it be like this:
B)
As stated in the beginning of this post I was trying to prove the product rule, and consequently set this up:
A) feels a bit suspicious, but if it works(in the case above) it would make my task very easy, and B) seems a little pointless...
I'm feeling a bit puzzled at the moment, and any clarification would be appreciated.
This is impossible because the left side has an "h" and the right side does not.
For any h f'(x) is approximately so that f'(x)h is approximately f(x+h)- f(x), f(x+h) is approximately f'(x)h+ f(x) so, yes, (which only says "f(x)= f(x)"!).or should it be like this:
B)
As stated in the beginning of this post I was trying to prove the product rule, and consequently set this up:
A) feels a bit suspicious, but if it works(in the case above) it would make my task very easy, and B) seems a little pointless...
I'm feeling a bit puzzled at the moment, and any clarification would be appreciated.
Ah, I've seen this one before and I remember that I found it very easy to follow when I was initially introduced to the product rule. Sadly, when trying to recreate it myself I always seem to forget to insert the -u(x+h)v+u(x+h)v. I think that I will grow out of this weird sort of "partial amnesia" eventually though.