Online Calculus tutoring, GM073019

Online Calculus tutoring, GM073019



yes sir I'm doing well I heard you got a pretty good score on the ACD oh yes sir I got a 30 yeah you know yet whether that get you into the school you're trying to get into the University Arbor yes sir that gives me 80% of beta basically pays for 80% uh-huh cool okay so I think we left off the chain rule right yes sir okay first of all do you need a little riff on the power rule product rule and quotient rule I think that might help okay let's start with this right here hold on a second let me turn my screen on sketchpad let's start all right so that is 2x what is cosine X that's right cuz it's neat uh-huh that is three right uh-huh all right yeah if you get it right I won't say that is one plus cot and X now the derivative of tangent is secant squared secant squared okay and it's definitely worth memorizing all six trig functions the remember e the only three you have to remember or the sine tangent and secant yes sir in other words once you've committed those to memory then you can always figure out the cosine the cotangent and the cosecant that's right three of you just to make sure just I'm gonna write the trig function you give me it's derivative yes sir all right that is the cosine of X secant squared and notice why I do it this is it that the secant x tan x uh-huh that's all you have to remember okay the other that's all yeah because the others you can figure out if you didn't remember if you didn't memorize this you can figure it out based on okay that is the negative sign of X right in other words all of these Co functions the cosine the cotangent and the cosecant or negative I know there's a negative sign there I know there's a negative sign there so that's right what's that derivative that is the negative is that the negative cosecant square hats whatever function of that is okay that's right and that is the negative cosecant cotangent that's the way you memorize commit those three and commit the method you're going to use to figure out the other three but I don't know might be easier to just memorize all six there is a lot of memorization in calculus for sure it's worse than a trick all right let's talk about the chain rule yes sir but let's not talk about how much of it you're a man okay so the chain rule it's a better way to write wife's wait Oh x squared plus one to the one-half power wow that's a better way because we're gonna use the power rule so square radical signs do not help at all in calculus anytime you see radical sign whether it's square root third root fifth root doesn't matter convert it to a exponential or a fractional exponent that's right okay so Y prime is what now this is a composite function first of all okay in other words it's of this form that's what a composite function is that's right f of X would be square root of x and G of X would be x squared plus one so if you were looking at what F of G of X is it's the square root of x squared plus one that's right okay and when you are doing derivatives you do the derivative of the outside first that's what that means and then you do the derivative of the inside and you multiply so if I'm taking the derivative of F of G of X I'm gonna do F prime of G of x times G prime of X that's where the verbal comes from where you take the derivative of the outside times the derivative of yes sir so what's the derivative of this thing all right so that's going to be let me see if I remember how to do this correctly so that heated up by pot a little bit above the inside that's boy always wanna be think 1/2 times x squared plus 1 plus hold on we have to do the exponent oh that's right my way it's the exponent minus minus 1 ok I might explain to the negative 1/2 plus x not blow but remember it's the derivative of the outside times the derivative of the inside yes so that is 2x yeah 2x that's the answer now dealing with a multiple choice test then this you might not see that as a choice because usually when they have a problem and they start you with a radical sign and your answers are going to have a radical sign so if I convert that to a radical sign 2x is in my numerator 2 difference that my denominator yes so that's the equivalent of what's up what it above it and in fact I can do that so this is the answer you would see on the multiple choice of course you could have done that to begin with awesome all right let's look at another one the derivative of the outside that is the cosine of 6x the derivative of n6 and no even though the me I'm always gonna remember it as derivative of the outside times the derivative at the inside in most cases you end up writing it the derivative of the inside times the derivative at the outside because you can kind of see why this is the better way to write it if I write it like this that's confusing I don't know if that's the whole argument of cosine or where the six is separate from the trig function that's right so you almost always want to put that number in front like that now what ends up happening is it usually you write it as the derivative of the inside then it's derivative of the outside the problem with doing it that way is is that frequently there are two insides so the rule becomes the derivative of the outside times the derivative of the first inside times the derivative of the second inside or times the derivative of the third inside and then you you know you simplify it after you get it all done how about this one here five times 3x plus two to the four times three your answer yes remember derivative of a sum or difference of two things is the same as the derivative of the first thing plus the derivative of the second thing when there's a multiplied not true when they're being multiplied yet is a product rule this was x times tangent of x squared yeah we'll do that after we do this all right so we're gonna have one plus the secant squared of x squared times two x now make that a multiplication just as further review of the product rule and the chain rule okay so the product Rock okay here let me write up here in the right you have two functions together the product rule is the first function times the derivative of the second plus the reverse that's what you want to remember it's really easy to remember if you think about it it's the derivative of one of them or it's one of them times the derivative of the other one plus the reverse that's which after them so I recommend everybody started out with that one whatever you is start your writing there all right so X now say it's a relative of tangent of x squared is secant squared x squared plus hold on I will apply that chain wait wait what yeah hello that's we're done we're doing V prime V is tangent of x squared so you have the full derivative of tangent of x squared and in that derivative apply the chain rule also that's right so times 2x correct and I can now multiply that produce that okay yes sir plus y plus it's just gonna be 1 so tangent of x squared all right that's the answer and I don't think I can do anything with that in terms of simplifying it ok now most chain rules are pretty obvious what is the inside and what is the outside there is one in particular that people have trouble with and for a reason you can understand this one or the subtleties of this one you'll have chain rule master what's y Prime and the cats are both of these you have to know what is the inside because that's what it comes down to is knowing what the inside is you can't take the derivative the outside times the derivative the inside and let you know at the inside do you want me to just use the chain rule for the top one or you use the chain rule for both of them but both of them require you to know what the inside is so yeah go ahead and apply it to the top okay so you first have a sine squared see inside the inside is X actually the inside is sine of X and the reason is here let me write it up here I'm gonna let u equal the inside so what we're looking at here is U squared all right oh and now if I apply the chain rule to that I'm gonna do to you times the derivative of U which is two sine x times cosine X with me yes sir I think so well I'll leave it up here I'm not gonna erase anything coz now let's do the second one what's the second inside so this is x squared Y a completely different so if y equals the sine of U what is y Prime that means y prime is the cosine of u times d u DX what is d u DX that is take the derivative of u 2x yeah now if I go back and substitute this is actually x squared so the derivative of the outside is cosine of x squared times the derivative of the inside which is 2x yes sir okay and like I said if you can understand the subtleties of these two problems you're 90% there in terms of the chain rule the difference is between these two yes as everything is what is in the inside the first case the inside is sine X the second case the inside is x squared so the formula for multiplication is you the inside or the outside the inside the inside yeah so this first one is actually u squared whereas the second one is the sine of u they're so raised to do this first one I can make it be sine of X times sine of X and use the product rule but that would not be the best way to do it because this subtlety between these two questions is the the biggest part of the difficulty of chain rule is you have to know what the insiders okay so most of the time it's a lot clearer than it is here okay we had one earlier that was tangent of x squared what's the inside of that it's x squared yeah so I meant that the derivative of the outside is the secant squared of U times the derivative of the inside 2x so knowing what u is is critical sometimes most of the times not mostly you don't have to get that deeply into it most of the times it's going to be clear this is probably the hardest case I can think of when I'm teaching people the chain rule is that use this as a big example because it didn't get much harder than that in other words if I give you something like this cube root of x to the fourth minus 2x plus 1 well that's easy what the inside and outside is of that right insurance if I convert it to this so what power the 1k give me the derivative of that that is one-third times X to the fourth minus 2x plus one to the negative 2/3 power x 3 x 2 1 for X there yeah well we start with that X love it don't start anywhere but that X 1 minus 2 that's it all right you see if I can find some difficult chain rule situations huh well I'm just gonna go through the ones on the book they're pretty interesting yes you now I understand the I haven't shown you anything practical about calculus yet you just got to get over that and get through all these memorizing things and then we'll we'll start talking about where it's applied in real life and how it's useful but for the first few weeks of calculus it's pretty much all rote memory stuff it's difficult to apply it to anything other than the slope of the tangent line and that is key always remember that that's what first derivative is is the slope of the tangent line because the next questions are going to be what is the equation of that tangent line when X equal 3 and okay so that's we'll start getting into more real-life problems there but until then it's pretty much all memory as the so we're going to use the quotient property awful lot of time you can change the quotient rule into the product rule by just rewriting it in fact that's the way I did it for the first six months I learned it because the quotient rule was so hard to memorize and I didn't have a nice convenient way of memorizing it so I changed all quotient rule problems for product rule so how would I write this to use the product rule instead of the quotient rule would you just that would be okay okay how's thinking okay yeah yes that's right and now apply the power rule along with the chain rule alright so that is negative 1 times X plus 1 to the negative 2 times 1 correct that is negative 1 over X plus 1 squared so whether the product rule or the quotient rule is easier here I'm always going with the product rule because that quotient rule is a devil to reduce means it's a you need all your algebra powers to reduce it whereas the product rule comes out pretty much in its final form immediately and looking for tough ones and they're getting hard fine rewrite it first so that is x squared minus one squared all to the one-third power amplifier okay so we're just simplifying this yeah in other words there's two different ways to do this problem one is I actually I actually have a composite function that is the result of three function notice I have the cube root function I've got x squared minus one and I have something squared those are the three functions so I could apply the chain rule twice and we'll get to that later but I don't actually need to do it here if I simplify it first I simplify it first then I only have to apply the chain rule once so let's just simplify patient of that so that's going to be x squared minus 1 to the 2/3 now take that derivative final annual once 2/3 times x squared minus 1 to the negative 1/3 power times 2x okay keep looking you usually don't have to look very far find tough problems in this book that I have it's good a book as there is as far as I'm concerned on calc this is the AP count hook which means college-level ok do we want to rewrite this for the product rule the next line all right so that's going to be negative 7 times 2t minus 3 to the negative 2 and so with the negative 7 always start right there almost always take multiple I would ever coefficient u hat oh that's right ok so it's going to be 14 times 2t minus 3 to the negative 3 times 2 just talk about this verse second all right is there an f of X that is undefined is there an X that makes f of X undefined this f of X mMmmm bad example yes no no that's a good example yeah in other words this is now in some cases in other words you could take the derivative of this easy enough right yes sir but the mere fact that this function is this function means that our constraints on what X convey in other words let's talk about the domain of X well he's at a negative 2/3 no it is I wrote it wrong sorry that actually matters hold on a second yeah let me change it straight a point here let's make it negative 2/3 all right now what is the domain of that function right there so the way to think about domain is for thinking about what X can't be cuz usually a lot of times don't mean is all real numbers and so thinking about it in a way we're thinking what the domain can't be what X can't be where do you come up for this it's oh okay now now it now I need something so the X cannot be one correct so X no negative one would be okay because negative one makes that denominator minus two squared which is four and the cube root of four is not zero so the only thing you got to be careful about with a lot of these domain questions is don't let the denominator be zero so start your problem out with that cannot be equal to zero so what does that mean it means X can't be equal to one that's right all right these problems it's gone strong Matt yeah we write it y equals x squared times 1 minus x squared to the one-half now you can't further do anything with that so what is y Prime so would we have to use multiplication for this well missus definitely the product rule because yes you have two daughters luncheons that are being multiplied yes all right so we have x squared times one half times one minus x squared to the negative one half times negative two x perfect um don't worry about simple it might be worth simplifying let's see what we get wait do we still have one more oh yes my fault so it's just it's just going to be plus two x times one minus x squared to the one of them yeah okay not quite on the know that is perfect yeah in other words you did u times V that old thing is V prime Plus u prime times V and drink of it so you don't make mistakes like almost it and I tell you that this is an easy one mistake to make where you drop it after you do the first part of it you forget the second part at least that's an easy mistake for me today is forgetting the rest of what's being added to that it's not the first person to make that mistake no I'm okay let me look some more so we have what's the inside the inside is 3x squared ok what's the derivative of the outside so the derivative of the outside that is the negative sine of 3x squared times 6x good so that is negative 6x sine 3x squared right yeah I mean I'll have to rewrite it as long as I use this print this is like I did then that is not the argument of the sine function yes but it's usually correct to rewrite it to avoid any difficulties hmm this is interesting it's to this one very similar this is really righted if I would not eliminate all ambiguities in other words if I want to make sure that 2x is not part of the argument I got to write it like this all right so this would mean that it's cosine of three times two plus the sine way negative sine of three times 2x just check to make sure we end up with the Clark answer hmm now I get why I didn't let's go back and do this huh sorry about this oh let's do that one is just x squared uh-huh all right so cosine of three times 2x so 2x plus out of three plus negative sine of 3 times x squared which gives that guy how do I not get this just a second must be some simplification I'm not seeing hmm just a second I mean they're looking at a typo or I'm looking at my own brain fart I don't get that answer back am i doing it now now I know what I'm doing yeah that's not the answer first of all what is that that is is that um it is it is it that the sign of something can't be greater than one no this has more to do with knowing whether that's a number or is that a very is it a number stick it into your calculator not cosine of 3 is a number okay okay and the derivative of a number is zero so this whole part of it gets crossed out and this becomes your answer because if the cosine of 3 is a number then the derivative of it is not minus sine of 3 that's actually what was wrong the derivative of it is serum me it's only if there's a variable and yet the derivative is minus sine Nisha it's the same with say natural log of 2 if you want to take the derivative of natural log of 2 natural log of 2 is a number so the derivative of it is zero it's not one-half let's talk about that as long as I mention it y equal natural log of X means Y prime equals one over X right yes sir but if y equals natural log of two that's a number that's not a variable anymore so y prime equals 0 not 1 over 2 okay ok so wherever you can convert these what are they called functions transcendental functions which are the trig functions and some other kinds of functions but wherever they convert to a number make sure you recognize it so you can make the derivative of that see okay yes I'm glad we did that one I might not realize talk about repeated applications of the chain rule all right now remember when I write y prime here what i really mean as what is dy d now hold on no I don't know I can write it it's just my problem with respecting other words I want to know what Y prime I want to know what dy DT is that's what I want dy DT I want to take the derivative relative to T naught X which the same as writing Y prime ok all right start with the derivative of the outside so that is the cosign that is 4 3 times sine square – 14 correct now what's the first inside the first inside is not the last I don't know that it matters but in other words you can either figure out the first one or the second one obviously one of them is 40 right that's right but let's go in order here we just did the outside times the first inside what's the first inside sign of 14 okay it's the derivative of that cosine 14 now take the derivative of the second inside fourth that's how you do it when you're applying the chain rule twice and there are some functions where you could apply the chain rule six times or 12 times whatever there's no limit that's how many times you apply but it always comes down and I think that's the reason they teach it the derivative of the outside times the derivative the inside isn't for when there's only an outside of em inside but it's for when there's an outside and multiple inside mm-hmm and the way you have to kind of think about it is start with the outside and then just start working your way down through all of the insides hmm and obviously we take that for multiply the 3 so that becomes a 12 and that's the answer 12 sine square root of 40 times cosine of 14 all right now good place to talk about near the end we'll get to our first they want the equation of the tangent line of this graph at the point PI comma 1 that is the quintessential question and derivative differential calculus is give me the equation of the tangent line when you're talking about equations of straight lines what's the most important thing me yeah absolutely if you have a straight line the slope is by far the very first thing you want to consider well what's slope well let me ask differently what's first derivative equal to so that so here I can make this a little clearer maybe this is that function graphed okay man this is PI so I want the slope of that line on what the equation of the slope well whenever you're talking about the equation of a straight line you're always first of all find the slope yes sir so what's F prime all right so that's going to be to cut what 2 cosine X right huh Plus what gloss all right so this is going to be the negative sine of 2x times 2 ok so negative 2 mu minus 2 sine X sine 2 in other words don't change that when you're doing the derivative of the outside that's the first thing you did was you're gonna take the derivative of the outside and the derivative of cosine of 2x is negative sine of 2x hi so now the next step let me erase that graph right is they want what is the slope when X is PI so what is f of Pi in other words F prime evaluated when X is PI I don't really care what Y is everything's a function of X only thing I need to know is X is PI that's right so we just plug in PI for X into the derivative equation alright so that's going to go sine of 5 minus 2 sine of 2 pi that's right let's come up with that number what's 2 cosine of PI equal to that negative 2 yep and what's sine of 2 pi equal to that is 0 so our answer minus 2 that is the slope of the tangent line when X equal ha ok now there are two standard equations of straight lines one is y equals MX plus B that's the one you almost never use in calculus the one you always use in calculus it is point-slope okay which is this one right here it's called point-slope cuz you have a slope and you have a point okay well we have a slope and we have a point so let's put it into slope point format what slice of water y sub 1 is that's 1 right correct what's the slope negative 2 thesa X minus pi and that is the final answer I do not have to simplify that it is in point-slope form that and that's always the format you deal with in calculus because what you usually have is one point and the slope the slope being derived from the first derivative so this is always what your answers are going to look like and you don't simplify it I mean if it's a multiple choice you may be looking at you know y equals something but in general you do not have to go beyond that's a better answer than actually y equals some just because that's the way of calculus is to put it into point-slope form that so let's do another one of those these are useful and they are so needed I guess I'm gonna have to make some progress maybe not yeah here I guess so let's do this the instructions on this are come up with the equation of the tangent line at that point all right rewrite it so that's three x squared minus two to the one-half power so that's one half times three x squared minus two to the negative one half times 6x now what's the next thing I need to write are we going to simplify that no yeah no need to because this is a middle step of the answer they're not asking us for F prime of X they're asking us for the equation of the tangent line for our own self we need to figure out what f prime of X is but we don't need to simplify so we'll just do we need to evaluate it at what number three correct now notice that why waste I'm simplified that's right okay now what is the slope at this point 3 comma 5 is it um it's in the numerator where's the numerator is made up of a few things so it's not okay what's in the denominator that is 7 the square meter set yet square root of 7 where it is now it's also not super important I guess it may be let's go ahead and rationalize now be honest with you I don't know what they do knowing root 7 over 7 okay now what's the equation of the line here let me make just a little bit of room here and we figured out the slope we have a number for the slope well for men are we gonna do is to put it into the equation of a line what slow substitute everything y minus 5 equals 907 over 7 times X minus 3 yeah yeah that is the equation of that tangent line yes sir and like I said if you were to take a test and make it only ask you one question that's what it would be give me the equation of the tangent line of this function at this point yes sir well this is definitely something you want to remember how to do all right that's a good place to stop to look a little bit in advance okay you know it looks like the next thing in the book is implicit differentiation it's a good topic all right garrison I'll talk to you next week yes sir see you then you thank you bye bye have a good one

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