all right let's take a look at the practice test for the instructions we're asked to find the difference quotient all right for the first problem we're given f ofx is equal to 5x - 3 again we're trying to find the difference quotient so I have that written out Force already again this is f of x + H - F ofx over H so what I'm going to do is just come back up here and figure out what is f ofx + H first again if you want to just jump right into the formula you can but I find this to be a little bit helpful if you're struggling so this is equal to again everywhere there's an X in this case there's just one you're going to plug in that X Plus H so let's say this is 5 * the quantity x + H and then minus 3 I make this three better here and we're just going to distribute this in to each guy so this would be equal to 5 * X which is just 5 X plus 5 * H which is just 5 H and then minus 3 let me get rid of these middle steps here and let me just move this up and let me copy F ofx real quick so f ofx is equal to 5x - 3 so these are the two things we need let me go ahead and grab this real quick and I'm just going to come down here to where we have our formula so basically I'm going to say this is equal to your f of x plus h which is this guy right here well that's just going to be 5x + 5 hus 3 so I'm just going to erase this now you don't need it anymore and then you're subtracting away F ofx now you got to be real careful here let me move this down so it fits you got to be real careful here because you're subtracting the whole thing away so you're going to go minus and you're going to open up some parentheses and you're going to throw this in here so the Quan 5x - 3 when people make a mistake on this problem this is usually where it's coming from just not paying attention to the signs so you want to put a minus out in front of some parentheses and let's put this over your H and so now you're going to distribute that negative into each guy again I like to go plus Nega 1 here to make that obvious so this will end up being 5x + 5H - 3 and then you have- 1 * 5x that's -5x and then -1 * -3 that's going to be + 3 and this is all over H so a lot of stuff's going to cancel here you see that you have 5x and 5x you see that you have -3 and positive3 so what you're left with here is going to be 5 H over H and when you look at that again you can do some cancelling so H over H that's going to be one so you end up with just five as your answer all right for problem two we have f ofx is equal to 3x - 7 and again we want to find the difference quotient so this guy right here so we're want to start with finding f of x plus h so what is f of x + H well this is three times let me make this three better again where there's an X you're just plugging in this X Plus H that's all you're doing so this is the quantity x + H here and then minus 7 so now we're going to use our distributive property so we're going to multiply this three by this X and then by H so let's say that this is 3 * X which is 3x X and then plus we're going to have 3 * H which is just 3 H and then you have - 7 so you can get rid of this and just move this up here like that and then of course I'm just going to copy f ofx is equal to 3x - 7 so let me grab these and come down here to where we have our formula and let's just go equals we can keep this simple I'm just going to erase this this is your f ofx plus h so we're going to put this out in front so this is going to go right there and then we're subtracting away again your f ofx so I'm subtracting this whole thing away so you want to make sure you have parentheses here again when people make a mistake that's where it's coming from so let me move this up here and let me get rid of this we don't need this anymore and I'm just going to close this down so then you want to put this all over H so we're going to distribute this negative into each term again you can go plus1 and just do it like that I think that's the easiest way so this is going to end up being you have 3x + 3 H - 7 and then -1 * 3x is -3x you have 1 * 7 that's + 7 this is all over H again you're going to see a lot of stuff canceling here so you have 3x and -3x then you have your -7 and 7 so7 + 7 would be zero so that's all gone all you're left with here is just 3 H so 3 H in the numerator over H in the denominator and the H's are going to cancel away and so what we're left with here is just going to be three all right for problem three we have f ofx is equal to 3x^2 - 5x Min - one so now we're getting into problems that are more tedious again coming down here I would just get the f of x plus h first so let me come back up here so f of x + H so this right here again everywhere you see an X you're going to plug in the x + H so this is going to be 3 * the quantity x + h^ 2ar then minus 5 * the quantity x + H and then minus one so let me come down here a little bit and again what you want to do is expand this so don't make the mistake of saying that this is x^2 + h^2 we know that we have to expand this and we're going to say this is three times so inside of parenthesis the first guy squared so X squ plus you need two times this guy times this guy so 2xh 2xh then plus the last guy squared so H squar so close that down and then this right here let's treat this as A5 times this quantity so I'm just going to use my distributive property and say this is5 * X that's - 5x and then you would have 5 * H that's - 5 H and then -1 okay let's keep going here so we're going to distribute the three into each guy inside the parenthesis so let's go ahead and do that real quick so 3 * x^2 is just 3 x^2 and then 3 * 2 xh is going to be + 6 xh and then 3 * h^ 2 is going to be plus 3 h^ 2 and then you have - 5x - 5 H -1 so so I realize this is long but this is what we have here so let me go ahead and grab this and let me come down here so this is going to be for the f ofx plus h so let me go ahead and paste that in and just put that right here so now we need to subtract away F ofx so let me come back up here and I'm just going to write this out so F ofx again is equal to 3x^2 - 5x - 1 so we'll grab this come back down and paste this in so let me get rid of this right here we're going to be subtracting this whole thing away so again you want to make sure you use some parentheses let me see if this is going to fit or if I need a new line Looks like it'll fit so let me move this up like that and just wrap it in parenthesis to make sure you're subtracting the whole thing away so this whole thing is going to be over H so let's say this equals we'll slide down here a little bit we have our 3x^2 + 6 xh + 3 h^ 2 - 5x - 5H and then Min -1 so this negative needs to be distributed to each guy so I'm going to go plus1 and just distribute this to each term inside of the parentheses here so -1 * 3x^2 is - 3x^2 -1 * 5x is + 5x and then 1 * 1 is + 1 so this right here is all over H well I see that I have 3x^2 and- 3x^2 so those are going to cancel then I have 5x and 5x so those are going to cancel I have ne- 1 and positive 1 those are going to cancel so it looks like what I have here that's left is this let me put equals 6 xh plus 3 h^ 2ar - 5H over H now you're going to notice that there's a common factor of H in the numerator so you got it here you've got it here there's just going to be one that you're going to pull out because this is squared and then you've got it here so what I'm going to do is I'm going to factor that out from the numerator and that's going to allow me to cancel out that H in the denominator so we're going to have an H out in front and so inside you're going to have 6X + 3 H - 5 and this is over H and of course this is going to cancel with this and so now what you have is 6X + 3 H - 5 and that's going to be our answer all right for problem four we have F ofx = x^2 - 25 again we have this right here so let me start with the f of x + H so f of x + H is equal to again I'm just going to plug in X Plus H where there's an X so this is the quantity x + H squared then minus 25 again we know we need to expand this so we're going to say using our formula this is X2 plus 2 * this guy * this guy so 2xh Plus plus this guy squared so that's h^ squar and make this plus sign better here and then we have our minus 25 so we can just copy this and let me put equals here and I'm just going to drop this in so this is the first part this is your f of x plus h now we want to subtract away let me make this equal sign better this F ofx so let me go up and get that so that is x^2 - 25 I'm not going to write it down I think we can remember that so let me come down here again you want parentheses because you need to subtract away the whole thing so this is x^2 - 25 close that down and then put this all over H all right now what I'm going to do again is just put plus1 right here and distribute this into each guy inside of the parentheses so let's say that this is equal to you have x^2 + 2 xh + h^ 2 - 25 and then1 * x^2 is - x^2 and then1 * -25 is Plus + 25 and this is all over H when we look at things again A lot's going to cancel so you have x^2 and x^2 you have - 255 and positive 255 and what you're left with let's say this is equal to 2xh + h^ 2 over H again you'll notice that you have an H here and you have an H here of course this is H squ but you have an H here as well so you can go ahead and factor that out so we'll say this is equal to so we'll have H * the quantity 2x + H and this would be over H and now we can cancel this with this and we'll say that our answer here is going to be 2x + H all right for problem five we're given f ofx is equal to X Cub + 1 coming down here again we're just going to start with the F ofx + H let's come up here let's say I want F ofx + H so this equals again where there's an X you're just going to plug in this X Plus H so X+ H this quantity is cubed and then plus one so this is something you would use a formula for let me go ahead and grab this real quick and let me come down here I already have this written out for us so we talked about this formula earlier on in the course some of you will remember it for some of you you won't remember it but it goes like this we have the quantity a plus B Cubed this equals a cub + 3 a 2 b + 3 a b^ 2 plus b Cub so basically all you're going to do is everywhere there's an a you're going to plug in an X and everywhere there's a b you're going to plug in an H because here a is first here x is first here B is second here H is second so you're just going to follow that formula so let's say this is equal to instead of a cubed you'd have X cubed and then plus you'd have three and then for a squ that's X2 and then for B that's going to be H then Plus plus you have three instead of a you're going to use x instead of b^ 2 you're going to have h 2 and then Plus instead of B Cubed you're going to have H cubed now you have this plus one here so don't forget about that let's put plus one in a different color so that's crystal clear I'm just going to grab this and come up here and put equals and I'll just paste this in right here so this is the f ofx plus h part we want to subtract away the F ofx again in case you forgot f ofx is equal to X Cub + 1 so let me wrap that in some parentheses so this is X Cub + 1 and then we're going to put this all over H so let me put equals here and then I'm going to write X Cub + 3x^2 H + 3 xh^ 2 + H cubed + 1 and then again let's go plus1 and just distribute this to each guy inside the parentheses so -1 * X Cub is - x cubed and then then 1 * 1 is -1 okay so this is all going to be over H let me make this X look a little bit better here and so if I go through and look you see a lot of stuff's going to cancel so you have X cubed and negative X cubed you're going to have this positive 1 and ne1 so what you have here is this 3 x^2 H+ 3 xh^ 2 + H cubed over H now what you'll notice is that you have an H here you have H squ here but you have an h and then you have H cubed here but again you have an H so I can pull out an H from the numerator and when I pull that out I'll be able to cancel it with that H in the denominator so let's go ahead and write this as H * the quantity this would be if I pull out an H 3x^2 and then plus if I'm pulling out an H here well this would be 3xh so 3x H and then plus if I pull out an H from here now this would be H squared so this would be over H like that so let me go ahead and cancel this H with this H and now we're going to have the difference quotient so here's where we'll get the answer let's put equals here and we'll just say this is going to be 3x^2 + 3x H + H squared all right let's take a look at problem 6 so we have f ofx is equal to -2X Cub - x - 7 so again when we look at our difference quotient we know we need f of x plus h so let's just start with that so let's do f ofx plus h so this is equal to again everywhere where there's an X so here and here you're plugging in that X Plus H so let's go -2 * the quantity x + H cubed and then minus you have the quantity x + H and then you have minus 7 so let's just bring this down here and paste this in so again we have this quantity a plus b being cubed as being equal to a cub + 3 a 2 B+ 3 a^ 2 + B Cub all you have to do is replace the a in this formula with X and the b in this formula with h that's it because a is the first guy here x is the first guy here B is the second guy here H is the second guy here that's all you have to do so let's say this is equal to we're going to put -2 times inside a parentheses again we're just going to match things up so instead of a cubed you're going to have X cubed then plus you're can have your three and then instead of a squar you'd have X squar instead of B you'd have H then plus you can have three instead of a you're going to have X instead of b^2 you're going to have H SAR and then Plus for your B Cubed again that's going to be H cubed close that down and then you might as well distribute this negative in now so let's go plus1 and just distribute that in so this would be Min - x and then minus H and then youd have minus 7 okay so let's say this equals this is going to be distributed to each guy so all of of these guys all four of these terms inside of the parentheses so -2 * X Cub is -2X cubed let me make this three better here then you're going to have -2 * 3x^2 H that is6 x^2 H then -2 * 3 xh^ 2 would be -6x h^ 2 then -2 * H cubed would be -2 h Cub then you have your - x - H and then - 7 okay let me go ahead and grab this and let's come down here and let's just paste this in so this is going to be for the f of x plus h so we want to subtract away this F ofx again you want to do this inside a parentheses let me go back up and grab this I'm going to write it real small so we can fit it so -2X cubed and then - x and then - 7 I make this seven better here come down here and paste this in and let me put that right there and then close that down so this would all be over H so what I would do here first is go plus again negative 1 distribute that into each guy inside so you got three terms there so essentially let's say this is equal to -2X Cub - 6 x^2 H - 6 xh^ 2 - 2 h Cub - x - H - 7 let's make that H better there and then -1 * -2X Cub is+ 2x cubed -1 * X is + X-1 * -7 is + 7 then this would all be over H now let me see what we can cancel here so looking through we see that we have -2X cub and we have 2X Cub we see we havex and plus X we have --7 and + 7 so what do we have left we have this right here and then we have this right here so don't miss that and what we're going to do is just factor out an H I should probably rewrite it first so it's more clear so let me write this is equal to -6x 2 H - 6 x h^ 2 - 2 H cubed and then - H so this would all be over H again you have an H everywhere so here this is H s this is H Cub this is H so we're going to pull that out and then we're going to cancel with that H in the denominator let me slide down here and say this is equal to pull out the H inside you have - 6x^ 2 - 6 x h - 2 h^ squared and then minus one again because I'm pulling out an H make sure you put a one there if you need Clarity you can always go times one there just so that's easy to understand so this would all be over your H there and again then this would cancel so now that's going to leave me with my difference quotient so this is what I'm looking for so -6 x^2 - 6xh - 2 h^ 2 - 1 let me fix this x here and this is going to be our answer all right for problem s we're given f ofx is equal to 7 over x -1 so once again we want to find this f of x plus h first to get things started so let me write out f of x + H here so this equals we have 7 over in the place of X you just want x + H that's all it is then minus one let's say this equals so I'm going to just paste that in like this hopefully you can remember that F ofx was just 7x -1 again if you can't we just come back up here f of x again is just going to be 7x -1 so I'm just going to put minus you have 7 over X - 1 then this whole thing is over H now this is a complex fraction there's lots of ways to work with it what I would recommend is just multiplying the numerator and denominator of the complex fraction by the LCD so the LCD here if you look you have a denominator of x + hus one and a denominator of x - one so it's just the product of those two so let me put times here and I'm just going to wrap this for clarity just so it's clear what's going on so this is times you're going to have this quantity x + h -1 * thean x -1 over again thean x + H -1 * the quantity x - one so again this by definition is going to be one it's just a complex version of that and so I'm multiplying the numerator of the complex fraction by this and the denominator of the complex fraction by this in order to simplify and let's put equals so this is going to get multiplied by this let me write it all out and then I'll cancel some things and make it a little bit shorter so 7 overx + H -1 let me wrap that and I'm just going to say this is times the quantity x + H -1 * the quantity x -1 so what's going to happen is this is going to cancel with this and you're left with 7 * this quty x -1 so let's just do that as 7 * quty x -1 and you can distribute that now or you can wait I'm just going to wait I'm going to leave it in that form for right now then you would have minus you have this 7 / x -1 again let me wrap this times you have this quantity x + H -1 * this quantity X - 1 so now this is going to cancel with this and you're going to have this seven times this quantity so let me get rid of this and again I'll distribute this later let me move this up here like this and this will be over now you're going to have H times this right here now I would not distribute this when you look at these problems you realize that your goal is to cancel the H in the denominator so just leave it out in front so I'm just going to say this is H * this quantity x + H -1 * this quantity X - 1 so that's all you need to do so let me slide down here a little bit and let's just simplify so I'm going to distribute the seven into each term inside of the parentheses so you'd have 7 * X which is 7 x and then- 7 * 1 which is 7 and then you're subtracting this whole thing away so be very careful I would put some brackets here to make that crystal clear just so you don't make a sign mistake so let's put minus you're going to distribute this into each guy inside the parentheses let me put my brackets there and so 7 * X is 7 x then you have plus 7 * H is 7 H and then you have minus 7 * 1 is 7 now let's close that down and again this needs to be distributed to each guy so you can go plus Nega 1 here just to make that Crystal Clear in terms of what's going on you're just going to change the sign of everything so we can do this right now we don't need another line so let's just go ahead and say that that this is going to end up being a negative this will be a negative and this will be a positive so let's get rid of all this just change the signs of everything so again this is going to be a negative this is a negative and this is a positive so let's put this over we have our H * the quantity x + H -1 let's make that a little bit better there and then times the quantity x -1 let's slide down here a little bit and let's look at what we can cancel so you have 7x and -7x you have -7 and 7 so what you have left is this -7 H so what you're trying to do is get rid of that H from the denominator so what you're going to say is this is equal to7 H over H * the quantity x + H -1 * quty X - 1 make this better here and I'm just going to cancel this with this and so now we're going to have our final answer here so we're just going to end up with -7 over you have this quantity x + H -1 * the quantity x -1 and I would just leave it like that so this is the difference quotient all right for problem 8 we have f ofx is equal to 4x overx - 5 and again we're trying to figure out this difference quotient here so let's start with f of x + H so let me slide down here so if we want f of x + H this would be equal to well you're going to have an X here and also here so you're going to say this is four * the quantity x + H over you're G to have x + H minus 5 so this I would distribute in so you'd have 4 * X which is 4X plus 4 * H which is 4 H so this is 4x + 4 H so 4 x + 4 H let me go ahead and grab this let's come down here and paste this in right there then we're going to subtract away again F ofx if we come back up is 4X over x - 5 so we want 4X over x - 5 again this whole thing is going to be over H so what you're going to do just like in the last example you're going to look at the denominator here and you're going to look at the denominator here and again you're going to multiply the numerator and denominator of this complex fraction by the least common denominator which is the product of these two denominators so for clarity let me just go ahead and wrap this and wrap this and we're going to multiply by the quantity x + H - 5 * the quantity x - five so we're going to do that to the numerator and the denominator let me make these better here and slide down here a little bit and let's see what we have again I'm just going to take this piece by piece so this equals I'm just going to write this out so 4x + 4 H over you have x + H - 5 so this is times you have this quantity x + H minus 5 time the quantity x - 5 so this right here is going to cancel with this right here so what you're left with is the quantity 4x + 4 H times the quantity x - 5 so we will do that later don't worry about it for right now let's just slide this over here like that and again don't worry about it for right now so then we're going to go minus we're going to have this multiplied by this so you have your 4X over x - 5 again multiplied by this so times your quanti x + H - 5 * your Quan x - 5 let me wrap this for clarity and we're going to cancel this with this and so now you have 4X times this quantity here again we're not going to do that just yet leave it for a moment so let me move this over here like that and then let me put this all over now you have H times this so H * the quantity x + H - 5 * the quantity X x - 5 okay let's go through and do some Distributing here so let's start with the 4X * X that's 4x^2 then we're going to do 4X * -5 that's - 20x then 4 H * X that's + 4xh and then 4 H * 5 that's going to be - 20h then again you're subtracting this whole thing away you can put brackets there to make that Crystal Clear let's go minus we'll put our brackets in here 4X * X is 4x^ 2 and then we have plus 4X * H that's 4X H and then we have minus 4X * 5 that's going to be 20x so again if you're removing this just change the sign of everything so I'm going to drop this and so this is a minus this is a minus and this is going to be a plus don't forget to do that otherwise you will get the wrong answer so let me go ahead and move this down here and let's put this over we have H * the quantity x + H - 5 * the quantity x - 5 okay so looking through what we have here you have 4x^2 and 4x^2 you have -2X and 20x then we can also cancel this 4xh with - 4xh and what we're going to be left with is just going to be this -2h over H * the quantity x + H - 5 * the quantity X - 5 me extend this a little bit so you'll notice that this H would cancel with this H and sliding down here now we're going to have our difference quotient so let's say this is -20 over you have the quany x + H - 5 * the quantity x - 5 all right for problem 9 we have f ofx is equal to the Square < t of x - 4 again coming down here and looking at the difference quotient we know we need this f of x plus h so let's start with that so going back up what is f ofx + H here again where there's an X you're just going to plug in X Plus H so this is the square root of x + H and then you have minus 4 so let's just grab this we can remember that f ofx equals the < TK of x - 4 no need to come back here so let's go equals I'm just going to paste this in so again this is the f of x plus h and then minus for f ofx it's just the square root of your xus 4 so this would all be over H now with this particular problem you have to take an additional step and rationalize the numerator because you want to get rid of H from the denominator so I talked about this in the lesson so basically you just want to remember if you have something like let's say A + B so this quantity times the quantity a minus B this equals a^ 2us b^2 so this is something we learned very early on in the of course this is multiplying conjugates and that's the formula so basically you're going to treat this as though you have this a minus B here so this is your a and this is your B what you'd want to do is multiply by you're going to keep this the same and this the same you're just changing the sign so you want the square root of your x + H and then minus 4 and then instead of a minus again you want a plus and then you want the square root of x - 4 let me get rid of this we don't need it anymore this would be over again to make this legal you would say the square root of x + H - 4 and then plus the square < TK of x - 4 you can just use your formula you can use foil too if you want let me wrap this stuff so this is Crystal Clear what's going on but you can go through and use foil just realize that when you do the outer and the inner that's going to cancel away so all you really have to do here is take this first guy and square it so we're just going to say that's x + H - 4 then we're going to go minus so here's where you have to be really really careful you're subtracting away this guy being squared so that's going to give me this xus 4 again I've got to subtract the whole thing away that's why I'm putting a minus in front of these parentheses again I can distribute this now just get it out of the way and go plus1 here and distribute this in usually when someone makes a mistake it's because they forget to wrap that in parentheses so they make a sign mistake so make sure you do that so this becomes x + 4 so let me write - x + 4 and this is all over again my goal is to get rid of the H in the denominator so don't distribute it in just write H * the quantity the square root of x + H - 4 and then plus the Square t of x - 4 close that down like that let me extend this a little bit and then all we're going to do at this point to finish this up is look at this canceling with this so x - x is 0 and then you have -4 + 4 so that's gone and so you just have this H right here that's going to cancel with this H right here so let's put equals here and I'll just come down here and say that this is going to be H over we have the H * the quantity the square root of x + H - 4 and then plus the square < TK of x - 4 make that a little bit longer there and then let me close this down Okay so this H is now going to cancel with this H and I'm going to put a one up here and so let's say this is equal to this will be our difference quotient here so this will be 1 over we have the Square t of x + H - 4 and then plus the Square t of x - 4 so that's going to be our answer all right for problem 10 we have f ofx is equal to the < TK of 7 - x^2 again just to start off we have our difference quotient here and we know we're going to need f of x plus h so let's put our f of x + H here and this would be equal to the Square t of 7 minus in the place of X you want to plug in the X Plus H so we want X+ H as a quantity here and this is going to be squared so be really really careful here because you are subtracting this guy away so if you're going to simplify keep that in mind but right now I'm just going to leave this as it is we're going to end up simplifying it later I'm just going to copy this real quick I just want you to remember that your f ofx is equal to the square < TK of 7 - x^2 just so we don't have to come back up and so we'll come down here and say this is equal to I'm just going to paste this in so this is for my f ofx plus h and then minus again for the F ofx we just said that's the square root of 7 - x^2 and this would all be over this H right here so again with this type of problem just like we saw in the last problem and also in the lesson we know we want to rationalize the numerator because we want to get rid of the this H in the denominator so we're going to multiply here by the square < T of 7 minus the quantity x + h^ 2ar and then what's going to happen is this sign is going to change instead of a minus you want a plus and then you want this guy right here so you have the Square t of 7 - x^2 this would all be over again you're just going to copy this so you want the square root of 7 minus again the quantity x + 8 h^ 2 and then plus the < TK of 7 - x^2 so all I'm really doing here is multiplying by a complex form of one so I'm not really changing anything I'm just going to change the way it looks and this is going to allow me to eliminate H again for the formula here just like we talked about in the last example this is really just a + b * a minus B so this quantity * this quantity this gives you a^ 2us b^ 2 so this is again known as multiplying conjugates so here this would be your a so this one and this one those are going to match this is your B so this one and this one those are going to match and notice that the sign here and the sign here those are different so when you go through and you crank this out all you're going to do again you don't have to use foil here because the outer and the inner those are going to cancel so really all you have to do is just take this guy right here and square it so that's just going to give you the right hand so we're going to write 7 minus the quantity x + h² weird then here's where you have to be very careful it's very easy to make a sign mistake you are going to subtract away this guy time this guy so in other words I'm just going to square this and I'm going to get my radicand so I'm going to put some parentheses there because I want the whole thing subtracted away so 7 minus x^2 close that down then this would be over I'm going to have H times the quantity so this guy's going to go in there so you have your square root of 7 minus the x + 8 h^ 2ar and then we'll have plus our Square < TK of 7 - x^ 2 me make this a little bit longer let me close this down like that okay so now this is where I would simplify this right here again down here you're not going to have to worry about it I would just leave it in that form so let's say this is equal to this is 7 minus this guy right here I would use some brackets or parentheses whatever you want let's just use brackets so if you expand this again it's X2 and then plus 2 2 * x * H and then plus h being squared so close that down and then this minus has to be distributed again you can go plus negative 1 like that and just distribute it to each guy let me make this better here so it looks a little bit cleaner and so this would give me-1 * 7 is -7 and then -1 * x^2 that's going to be + x^2 so you can do the same thing here and go plus negative one and then distribute that to each guy inside just like that let me put this all over your H * the quantity the square root of 7 minus the quantity x + h^ 2ar and then plus the Square t of 7 - x^ 2 and let me close that down so I'm going to write this result here on another line let me come down here and so this would be seven and then again this negative 1's multiplying by each guy so 1 * x^2 isx2 -1 * 2xh is - 2xh and then -1 * h^ 2 that's - h^ 2 and then you have - 7 and then + x^ 2 so all over this H * the quantity you have the square root of 7 minus the Quant x + h^ 2ar let me make that a little bit longer and then plus you have your square root of 7 - x sared and let me close that down so in the numerator you're going have a lot of stuff cancel so you have this 7 and then 7 you have x^2 and x^2 and what you're going to be left with is this -2X H minus h^2 now notice you can factor out an H from that I'm just going to do that now just to speed this up just a little bit so let me pull out an h and inside you'd have -2X so -2X let me just highlight the H here and I know this is H squ but let me just highlight the H there so this is clear what I'm doing just pulling out an h and then basically you'd have minus you still have an H left there close that down this would be over again you have your H * the quantity this is the square root of 7 minus the quantity x + h^ 2ar and then you have plus the square Ro T of 7 - x squared let me just make this longer like that and let me close that down okay at this point now you can cancel this H with this H and we're ready to present our answer so I'll go ahead and say this is -2X minus h over I'm going to say this is the square root of 7us the quanti x + H squared so you could simplify that if you want I would just leave it as it is then I'd say plus this is the square Ro T of 7 - x^2 so that's going to be our difference quotient