00:01
So here we have a problem just given as f of x equals x to the 8.
00:07
There are a couple ways to interpret this, i'm going to interpret it in a couple different ways.
00:11
First, because this is just a calculus class, calculus exercise, we can try and find the derivative of x, which using the power rule is 8x to the 7th, you're going to get the 8 down in front and then take x to the 7th.
00:25
Or maybe we want to do this with the limit definition, so we can have the limit as h approaches 0 of x plus h to the 8th minus x to the 8th over h.
00:42
Well, this is going to be the limit of, and if we do expand out the binomial expansion, x to the 8th plus 7th, or sorry, not 7, that was a typo, plus 8h x to the 7th plus some other terms that all have h in them, minus x to the 8th, all this over h, the x to the 8th cancels out and we're left with the limit as h approaches 0 of 8x to the 7th plus h times some terms that if you foil out the 8 or use binomial expansion, all are going to include a factor, but i guess i should say at least a factor of h squared...