Calc Topic 2 8 2 9

Added: 2026-05-06

[00:00:06]okay topic 2.8 the product rule and topic 2.9 the quotient rule is what we're going to explore today so recall with the sum in differences you can take the derivative of each of them and then add and subtract them however that is not true for multiplication and division i'm not going to do any proofs um of these rules if you want to see the proof you can scan this code and watch the proof of the product rule we're just going to learn um how to do the product rule and how to memorize it so it says the product of two differentiable functions so if this function and this function are both differentiable then the product itself is differentiable so this is the product rule right here and basically what this says is if you have a first function times a second function the product rule to memorize is the first times the derivative of the second plus the second function times the derivative of the first so i wrote that up here for you so memorize the first function times the derivative of the second plus the second times the derivative of the first okay so we just have to memorize so let's go ahead and try this on h of x if you want to stop the video and try it on your own you can do that now okay so here we go the product rule the first times the derivative of the second plus the second times the derivative of the first and that's the product rule that's the calculus the rest is algebra now we have to simplify distribute the four foil and combine like terms x squared we have a negative 24 x squared combine the x's and the constants and that's your answer now do we have to use the product rule in example number one have we done problems like this in the past another option would be to foil this out first so we could start with h of x foil you get 15x plus 12x squared minus 10x squared minus 8x cubed so if we combine like terms that's negative 8x cubed plus 2x squared plus 15x so we could multiply them together and then use our power rule negative 24x squared plus 4x plus 15. and you have the same answer so i like this method better if you can avoid the product rule i would avoid the product rule but that's not always the case let's take a look at letter a you have to recognize that this is a product x times the sine of x so we have to use the product rule you want to pause the video and try the product rule for practice i would recommend okay so here we go f prime of x equals the first times the derivative of sine which is cosine you should have that memorized plus the second times the derivative of the first which is one so your answer is x cosine of x plus sine x go ahead and try b stop the video and try b and now check your answer first times the derivative of the second plus the second times the derivative of the first or you can also factor out an e to the x and have the answer of one plus x now these are multiple choice questions so you just have to make yours look like one of the choices either one of these is acceptable you don't need the one here if you don't want to write the one okay the next one i'm going to do is d so go ahead and stop the video and try d on your own okay let's check your answer the first times the derivative of the second you have to memorize that as one over x plus the second times the derivative of the first so simplify you have i'm going to put the 2x in the front just so we don't get confused with this product here 2x times the ln of x or another possible answer on the multiple choice could look like this okay all right let's try c this one's a little bit harder you have a two times an x times the cosine x many times we ignore the the two and then we take the derivative of this and then we multiply everything by 2.

[00:07:05]like over here we take the derivative of sine x and we multiply it by negative 2 because that's just a coefficient and this is a coefficient as well but the difference between this one and this one is we could use this as our first function and we could just we could just use it in our product rule which makes the problem easier than if you just multiply your product rule f with if you multiply your product rule by 2 after it's going to be a little bit more work so my suggestion here is if you can put your coefficient with one of your factors then this would be our first function and this would be our second function for the product rule and you have to recognize that that is a product okay this is a product you could use the product rule with two and sine x but the derivative of two is just going to be zero so you're wasting your time using the product rule when you have a coefficient you can just use our coefficient times the derivative of sine x so go ahead and stop the video and try letter c and now let's check your answer h prime of x equals the first times the derivative of cosine negative sine x are you okay with me putting that negative in the front first times the derivative of the second plus the second times the derivative of the first i'm just going to put that in the front now take the derivative of two sine x so we're going to multiply the derivative of sine x by negative 2. derivative of sine x is cosine x so again memorize derivative of cosine x is negative sine x the derivative of sine x is positive cosine x don't get those confused these two cancel out your final answer is negative 2x times the sine of x and that's the product rule okay let's move on to the quotient rule the quotient rule is the one that you want to avoid using at all cost because it's a lot to memorize but you still have to memorize it this is the quotient rule again if you want to see the proof just go ahead and scan that the quotient rule is the bottom times the derivative of the top minus the top times the derivative of the bottom all over the bottom squared okay so when i say top i mean numerator when i say bottom i mean denominator so memorize it however you would like but again it's here it's the denominator times the derivative of the numerator so i will say bottom times the derivative of the top minus the top times the derivative of the bottom all divided by the bottom squared so memorize repeating it over and over again so let's go ahead and find the derivative of y another notation for y prime is d y d x so you need to know that that is the derivative of y with respect to x so that's what we're going to do we have a quotient here so let's do that together so here we go the bottom times the derivative of the top minus the top times the derivative of the bottom and don't forget it's a common mistake don't forget your denominator all over the denominator squared and that is the quotient rule the rest is simplifying using algebra you want to go ahead and stop the video and simplify practicing your algebra skills be very careful this is a negative that needs to be distributed the other common mistake is people will switch these around and you can't do that make sure you follow the order so you don't make those common mistakes now you don't have to simplify your denominator you can just leave it so your final answer is negative 5x squared minus 4x minus 5 all over x squared minus 1 the quantity squared okay find the equation of a tangent line think what do i need you have to ask yourself what do i need to find the equation of a tangent line and you should say i need a point which is right here and what else a slope i need a point and i need a slope how do we find a slope what does slope mean in calculus tangent line slope in calculus what does that mean it means find the first derivative so we're going to find the first derivative of y again notation for the first derivative of y is d y d x or y prime and again we have a quotient here so we're going to use the quotient rule so why don't you practice the quotient rule on your own stop the video and see if you can do the quotient rule on your own okay let's check your answer bottom times the derivative of the top oops minus the top times the derivative of the bottom all over the bottom square memorize memorize memorize now on example number three we simplified it first i'm going to give you a suggestion to save you time if you're going to find a derivative and you're going to evaluate that derivative at some x value plug in the x value before simplifying and you will save yourself time in the long run so again this was just finding the derivative this we need to find the derivative evaluated at one so i'm not going to simplify this i'm just going to evaluate it at one let me show you the notation for that that you need to understand i'm going to evaluate the derivative i'm going to evaluate that at x equals one that's like saying what is f prime of 1 but we don't call this f of x they call it y so we're just using the notation that's given to us it's not in function notation so we're going to use this notation evaluate the derivative at x equals 1.

[00:15:59]so you have to understand that that's what that means so i'm going to go ahead and plug in 1 before i simplify so 1 plus one squared times e to the first minus e to the first times two times one all over one plus one squared the quantity squared all right so simplify that what does that equal to you should get an answer of zero over four which is just zero so think about what that means if i have a first derivative at x equals one equaling zero what does that tell you what does that tell you about the graph you need to know the first derivative equal to zero is where you have what and the answer is a horizontal tangent line so if i say find a horizontal tangent line you set your first derivative equal to zero if i say the first derivative is equal to zero you need to know that that's where the graph will have a horizontal tangent line so if the graph has a horizontal tangent line at this point we have a point and we have a slope so normally we would say okay let's use point slope y minus our y value equals our slope times x minus our x value now that was a free response question that would be an acceptable answer but you don't have to do that and that's not going to be the multiple choice answer if you have a horizontal tangent line at this point you should know what the equation is horizontal tangent line horizontal at this point has to be this and that would be the multiple choice answer if you simplified this you would end up getting this it would just take you longer to get there we're trying to save time so be smart okay so uh we've done these in the past recall when we did the power rule i told you that you cannot take the derivative of the top divided by the derivative of the bottom and you have to use the quotient rule so if you take the derivative of the top divided by the derivative of the bottom you're going to get the wrong answer so don't do that we want to avoid the quotient rule whenever we can so what i told you back in section 2 5 and 2 6 was that if you can get rid of any product like we did in example number one if you can get rid of any product or any quotient that would be the first thing that i would do because it's going to save you time you can use the quotient rule in each one of these but i would not time is of the essence so can you rewrite this so it's no longer a quotient monomial denominator monomial denominator do not use the quotient rule it just is going to take you longer so if we rewrite this 6 is the common denominator so i have x squared divided by 6.

[00:19:52]i prefer to write it as 1 6 x squared it's just my preference this is 3x divided by 6.

[00:20:00]i prefer that to write that as one half x so when i differentiate it i can see that i'm just using the power rule so 2 times 1 6 is one-third and then the power rule for this or the derivative of x is just one okay so that is your simplified answer for this one again i don't want to use product rule so i can just distribute negative 9x plus 6x squared over 7x and i don't want to use quotient rule so again let's go ahead and split it up rewrite it and again my preference is to write it on the side instead of writing it on the top it's just personal preference now take the derivative the derivative of a constant is zero and the derivative of x is one and that's times six sevenths again so that's your simplified answer okay now notice this derivative has an x in it and this derivative does not have an x in it what why is that well this is the derivative this is the slope formula think of it as the slope producing formula so if i wanted the slope of the tangent line at 1 i would plug in a 1 and that would give me the slope of the tangent line to this now look at this is a quadratic so as the x changes the slope of the tangent line is going to change so this is the slope the derivative of the slope of the tangent line the derivative of the function which produces the slope of the tangent line at any x this one doesn't have an x in it what does that mean that means this is going to be the slope of the tangent line for all x which means this must be a straight line isn't this a line isn't the slope six sevenths so every point on this line has the same slope has the same tangent line that's why the derivative is always equal to six over seven just something to think about all right let's look at some ap questions okay in ap they you cannot avoid using the product and quotient rule you can't avoid it on some problems like this one they don't tell us what f x and g of x are they just give us a bunch of information here and they tell us that f and g are differentiable and that's it so let's go ahead stop the video and find the derivative of h of x check your answer h prime of x is the first times the derivative of the second plus the second times the derivative of the first product rule okay now find h prime of three again stop the video at any time do it on your own for practice f of three times g prime of three plus g of three times f prime of three do you know how to read the table and find those values f of three is one g prime of three is negative four g of three is also negative 4 and f prime of 3 is 8.

[00:24:39]so that is a negative 4 plus negative 32 for a total of negative 36.

[00:24:46]now the applications of all of this will be coming in the next unit right now we're just practicing basic rules that you need in order to do the applications so that is coming okay let's take a look at b look at this notation a little bit different what does this mean this means find the derivative of this product find the derivative it's basically saying that y is equal to the square root of x times f of x that's what it's telling us that this is our y value and this is saying take the derivative of our y value with respect to x so it's asking you to take the derivative of our y with respect to x some people don't like how i jump from this how do i know that this means this again this is telling us that this is your y value your y function and they want you to find the derivative of the y function with respect to x so this is the proper notation d y d x is what you're finding so go ahead and try the product rule on your own check your answer the square root of x the first times the derivative of the second plus the second times the derivative of the first you should have memorized the square root rule one over two square roots of x now over here we wrote h prime of three well we can't really write d y d x prime of four that's not how we write it when it's in this notation so here is the notation that you need to understand again i did this before on the last page d y d x if you're evaluating it at four what would the notation look like d y d x evaluated at x equals four that's like saying h prime of 4 but with a different notation so you have to be familiar with all of the notations and what they mean okay so let's go ahead and evaluate that the square root of four times f prime of four plus f of four times one over two square roots of four so go ahead and find those values and see what you get stop the video at any time to work it out so these are the values you should come up with that's a negative two plus five halves which is positive one-half okay reading a table all right last one practicing the same thing with the quotient rule i would try these on your own turn off the video for a while try them on your own and come back and check your answers all right checking your answer h prime of x equals the bottom times the derivative of the top minus the top times the derivative of the bottom all over the bottom squared we're evaluating that at two so g of two times f prime of two minus f of two times g prime of 2 all over g of 2 squared so that is 3 times a negative 1. minus 4 times 3 halves divided by 3 squared which reduces to negative 9 over 9 which is equal to negative one got negative one good job if not check your numbers check your formula okay on b j prime of x the bottom times the derivative of the top minus the top times the derivative of the bottom again square root rule one over two square roots of x do not use the power rule this square root rule will save you time all over the denominator squared when you square a square root it goes away so we are finding evaluating that derivative at 4 the square root of 4 times g prime of 4 minus g of 4 all over 1 or times i'm sorry 2 square roots of 4.

[00:30:21]over four and for that you should have gotten two times one minus five times a quarter all divided by four which is two minus five quarters divided by four uh that is eight quarters minus five quarters so three quarters on the top divided by four is three sixteenths and there you are you