00:01
Find the difference quotient and simplify your answer.
00:04
We're given the function f of x equals x to the third plus 3x.
00:09
To find the difference quotient, we're going to use the form of x plus h minus f of x over h.
00:20
So what we want to do is plug in this part, the x plus h, and anywhere we see an x in the original function.
00:30
So we have x plus h to the third plus 3x or 3x plus h and then you want to subtract this part so we're going to just simply rewrite our original function because it's just an x or we're asked to plug in f of x.
00:52
So we have x to the third plus 3x divided by h.
00:57
So we can expand this first part into x squared plus 2 xh plus h squared we're going to multiply this by x plus h this is plus 3 x plus h and then let's just distribute this negative sign the subtraction sign to those terms and we get minus x to the third plus 3x we'll also distribute this in the next step.
01:31
So we're going to multiply these two together.
01:35
And we get x to the third plus 2x to the second h plus h or sorry x, h to the second.
01:53
And then distribute this h over.
01:59
And we get plus x squared h plus 2x squared h plus 2x h squared.
02:09
Plus h to the third.
02:13
Distribute the three to those terms.
02:15
We get 3x plus 3h minus x to the third plus 3x all over h.
02:24
So now we want to figure out which terms we can combine and which terms we can cancel out.
02:29
Oh, sorry, this is going to actually be a subtraction, or we didn't distribute negative sign here, so that's negative 3x.
02:41
So now we want to cancel out the terms that we can and combine the terms we can.
02:47
So we have a negative 3x and a positive 3x here.
02:51
We have a negative x to the third and a negative x to the third here.
02:55
And then we can combine the terms that have the same exponents on the x and h.
02:59
So we have this one and this one.
03:02
So we have going to keep 3x squared h adding the coefficients.
03:11
We have this one and this one.
03:14
So that's plus 3xh squared.
03:19
And then we have a plus h to the third divided by h.
03:35
And now we have an h in every term so we can cancel some things out.
03:38
We can cancel this h with this one, cancel this exponent, and drop this exponent to a 2, reducing the exponent on each h.
03:46
And we are left with 3x squared plus 3 .3.
03:50
X plus h to the second...