Tricky logarithmic differentiation example
Вставка
- Опубліковано 2 чер 2024
- ►Get my favorite calculator app for your phone or tablet: MAPLE CALCULATOR: www.maplesoft.com/products/ma...
►Check out MAPLE LEARN for your browser to do compute and graph math: www.maplesoft.com/products/le.... Thank you to MapleSoft for sponsoring today's video.
If you want to practice logarithmic differentiation, check out my worksheet here: ►learn.maplesoft.com/d/ASPMFLE...
This multiple choice problem was supposed to be a pretty easy example of logarithmic differentiation. 12% of students got it. Why is it so weirdly hard?
0:27 Basic Logarithmic Differentiation
3:27 The Hard Problem
6:26 Learning Mathematics
11:59 Teaching Mathematics
►Check out MapleSoft's UA-cam channel: / @maplesoft .
Check out my MATH MERCH line in collaboration with Beautiful Equations
►beautifulequations.net/pages/...
COURSE PLAYLISTS:
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►MULTIVARIABLE CALCULUS (Calc III): • Calculus III: Multivar...
►VECTOR CALCULUS (Calc IV) • Calculus IV: Vector Ca...
►DIFFERENTIAL EQUATIONS: • Ordinary Differential ...
►LAPLACE TRANSFORM: • Laplace Transforms and...
►GAME THEORY: • Game Theory
OTHER PLAYLISTS:
► Learning Math Series
• 5 Tips To Make Math Pr...
►Cool Math Series:
• Cool Math Series
BECOME A MEMBER:
►Join: / @drtrefor
MATH BOOKS I LOVE (affilliate link):
► www.amazon.com/shop/treforbazett
SOCIALS:
►Twitter (math based): / treforbazett
►Instagram (photography based): / treforphotography
Most students fail calculus because they are failing at the algebra. The calculus part isn’t too hard and easy to get. It’s the algebraic manipulations that get you.
Yep, this is exactly why I failed calc 2.
I completely understood the concept innately, but the algebra (especially when dealing with hard integrals, like trig subs and such) Is really a killer.
students get the equation wrong because capitalism teaches them to do things without understanding what they are doing. the sum rule is taught in the first year of university but if Western governments teach them to do things without looking at them, they get the equation wrong.
So true Sir
I’ve noticed a similar thing a lot when grading. Many students will apply the rote rules they’ve learned without really thinking about whether the rule is applicable. For example, trying to extend finite-dimensional concepts such as bases and linear combinations to infinite-dimensional spaces. Or not checking if a function has the required smoothness properties before taking a derivative.
*This.* As a math tutor, I always circle around back to one question: "Why did you do that?" - No, put down the eraser, I didn't say it was wrong to do it, I want to know *why you did it*. What about this problem made you decide that that rule or process was the thing you were supposed to apply at this point in time? Always always always take a moment, when looking at a new problem, to pause and examine it, identify what it is asking of you, what 'kind of thing' is it expecting you to give it as an answer, and make sure that you understand how what you're applying to the given problem helps you get closer to that answer; how it helps you break something complicated down into smaller, solvable parts.
Math is a language, yes, but Math is also a *toolbox*, a metaphor I come back to a lot. If you pull out the wrong tool for the job, you're not going to get the result you want - driving in a screw with a hammer is not ideal, even if it eventually 'works'. The process of learning Math is as much about learning *when* to use the tools in your toolbox as it is about learning *how*. The 'how' questions, the 'procedural fluency' testing, is very important; you can't use the tool if you don't know how after all. The question of 'when' to use the tool, however, and how to recognize not only the situation where it is useful, but how to recognize where in the problem to apply it, that is where so many students get tripped up.
This problem is sort of like recognizing you need a screwdriver, because the problem has screws in it, but trying to use it on the couple of nails in the problem as well, because "There are screws here, so I have to use the screwdriver to solve the problem."
I started solving all my exam questions before giving the exam -- though I usually keep the unintentionally harder problems and just put an * next to them to give the students the hint that it's harder than it looks, or sometimes I give a really big hint if I think the problem is likely to have a 0% success rate! z
Definitely nice to signal to students that there is something a bit tricky about the problem so they know to be on the look out.
I noticed something similar with integration of a sum, where one term needs a substitution but the other term doesn't. So now I always make sure to do examples of these type of problems.
Interesting observation.
In my school, problems like these were always taught to be solved through x^(f(x)) = e^(ln(x)*f(x)) substitution, and it completely skips the difficulty you've described, you can just use the standard substitution/summation/chain rule to solve it Maybe you can think of an example where this method would also fail, but I feel that it is in general more robust to errors.
Bless our 1st year calc teacher for teaching us this too
Ive got another idea about why this problem’s scores were so low.
It’s possible that the errors werent from careless mistakes, but from trying to get partial credit on the problem. After all, you did say many students that missed the problem did know the correct log rule.
The problem took me a few minutes to get, I immediately realized the same Log trick from before wasnt possible, but couldnt think of a way around the sum. Had I been taking this test, I’d still go through with that wrong log rule knowing it was wrong because you’re more likely to get partial credit on it. Its very possible that your students thought in the same way
That's definitely a possibility, they would have known they would get at least some points for that method.
I think this problem would have a higher success rate on a test following the former.
Another excellent commentary, Trefor; you made some really interesting points, and I'm curious to know more about mindful learning. One of the issues I find with my students is that throughout their primary and secondary education here in the U.S., the classroom focus has been teaching to the test. In other words, they can solve problems that are virtually identical to the ones they saw in class, but if there is any change or variation they are lost. They also seem more interested in learning "rules" rather than learning how to analyze a problem before attempting to solve it. I don't blame the students or their teachers, who are rewarded/punished based on the passing rates on standardized tests. But I think it's becoming increasingly clear that the "reforms" of the past 20-25 years have had some unintended negative consequences.
Honestly a big part is to not have learning objectives like the one from this video. That is, way more time playing/experimenting/exploring/making sense of mathematics, and less time on always trying to memorize the procedure to solve a defined problem.
Okay, huge relief felt when you said that you can do each piece separately. However it's tempting that the two pieces share things in common, almost like you could write one in terms of the other...
This is the difference between knowing something and grokking it. You can tell a student that differentiation and integration are linear operators until you're blue in the face but that has to be internalised before it's of any practical applicability.
Thanks for the great video. I'm going to keep this in mind when I'm setting up my course for next year :)
i never learned a "logarithmic derivative". instead, i found out after thinking about similar functions, that when
y=f(x)^g(x),
you can use the chain rule twice. (take derivative with respect to f and multiply by f', and also with respect to g and multiply by g')
y' = f^g ln(f)*g' + f^(g-1)*g*f'
y' = f^g (ln(f)*g' + g*f'/f)
when it comes to teaching, my goal is always to get the concepts across. but you can't overpress. often the results are better when you teach the recipies, and the student finds out about the general concepts by themselves.
Why even attempt to differentiate directly? Using the property (f +g)' = f' + g' is a natural approach.
I completely agree! That is exactly where I made most, if not all, of my mistakes in my Calc career. Simple algebra math or procedures were the death of me. I even got a bit stuck in this video and in the same moment you started explaining I said "Wait!!! Just take the derivative, there separated by a sum" 😅 Great video! Love it, you have accurately pinpointed the core issue I had going from pre to Calc 3. I wish I had found this sooner but all the more I've found it now!!
I think it may have to do with the notation used. Which in turn has to do with pattern recognition and the language of mathematics as you already covered.
When asked to differentiate y = x^2, it may hint that you are supposed to take logarithm of both sides, then differentiate both sides and you get:
y = x^2 => y' / y = 2 / x => y' = x^2 * 2 / x = 2x
If asked to differentiate f(x) = x^2, it hints that either you are allowed to say "hey, I already know this one" and directly put 2x, or possibly start doing the whole "(f(x+dx)-f(x))/dx as dx tends to zero"-thing.
The difference in notation is subtle, but one emphasizes a kind of graphical or geometric relation, and the other a function recipe relation. It is not at all obvious in an exam or test that the goal is 'to just solve the problem'.
I mean, I never learned "logarithmic differentiation". When I have to differentiante a function of the type y = g(x)^h(x), I just change it to e^(h(x)*ln(g(x)) and then take the derivative. Which makes the problem completely trivial.
In most case that method is completely equivalent and I'd call it the same thing.
I really find it helpful to restate the problem as a partial derivative of a multivariate function.
For example d/dx[(sin(x)+1)^(2x)^(x-1)] is hard to compute. But if we let
a(x)=sin(x)+1, b(x)=2x, c(x)=x-1,
g(x)=[b(x)]^[c(x)], f(x)=[a(x)]^[g(x)]; then the problem becomes easy.
d/dx[f(x)] = df/da•da/dx + df/dg•dg/dx
df/da = g(x)•[a(x)]^[g(x)-1]
da/dx = cos(x)
df/dg = ln(a(x))•[a(x)]^[g(x)]
dg/dx = dg/db•db/dx + dg/dc•dc/dx
dg/db = c(x)•[b(x)]^[c(x)-1]
db/dx = 2
dg/dc = ln(b(x))•[b(x)]^[c(x)]
dc/dx = 1
As you can see, instead of using logarithms and lengthy repetitions of product-power-chain rules, we simplified the problem into small steps. This way you only need to apply simple rules and combine the results together to get the solution.
Yes the multivariate chain rule makes finding all these complex derivatives very simple, and also means you don't have to remember any other rule since it implicitly gives you the product rule and quotient rule as well.
One of the problems that many people (like me) didn't solve in my last calculus 2 test was a limit of a function with a bunch of sines and cosines, where the function was actually continue and you could just calculate it. I felt so stupid when they corrected it.
That's tough. It'd be very unexpected and unacceptable if this were a multivariable calculus course where students are used to using properties of derivatives to solve problems and make proofs. In a first year calculus course it's very possible that the linearity of the derivative (or a sum rule) wouldn't be a students first intuition. Especially because first year calculus students are probably more mechanically used to being able to take a derivative of something that looks large and then applying chain rule and product rule to break it down. And in fact I imagine with a little time pressure it'd be easy to mistakenly apply the linearity step they know works for the derivative operator and mistakenly apply it to the logarithm. Sometimes early in ones mathematical maturity a hint on a problem can go a long way.
It isn't tough, students should just think harder
Makes me realise how much mathematics I have forgotten in the last 50 years.
NO! - let's be optimistic and assume you are teaching to a higher standard than I was exposed to back then, that makes it progress not senility.
Nice to see Maple being used.
6:41
Was this particular problem covered in class and assigned in the homework assignments ?
Was there an example of this problem work out in TA sessions ?
No! I mean, there are lots of logarithmic differentiation problems, but this specific issue of the sum there wasn't. That's a bit why I (and the students) were surprised, it wasn't clear that this detail was the big issue.
Pretty surprising that the vast majority of students would not first try to just differentiate the expression x^sin(x) = e^(ln(x).sin(x)), before attempting more advanced methods like the logarithmic differentiation!
Concerning logarithmic differentiation, could you do a tutorial for a function like: f(x) = (sinx+1)^x ? This function has an interesting domain, but also the derivative of f must be found by using the limit process at some points and to be fair we can't even use DLH rule.
That's quite a fun function actually!
If you rewrite the function as exp(x·ln(1 + sin(x))), then it becomes a lot more straightforward.
the short is just that learning things by rote reduces the amount of working memory required for a problem, and hard math problems are fundamentally about working memory. Mechanical fluency is UNDERRATED by liberal educators for that reason (who think that school success is mostly about effort rather than mostly about raw cognitive ability) b/c the kids who are successful all have a baseline level of working memory and so what differentiates THEM INTERNALLY is work ethic, which leads to the incorrect inference that it's work ethic that's more important than raw cognitive power (range restriction means they don't see all the kids who work hard but still fail, and aren't sensitive to how different kids have to spend different amounts of time on the homework--every student knows that smart kids spend less time on their homework, and to THEM the marginal hour really matters, but the educator isn't very sensitive to this, if they can even tell. But because of the different utility functions for kids time vis a vis kid's grades (who's a perfectionist, who likes to play videogames too much, etc, cognitive ability isn't correlated with GRADES). This is ironically regressive b/c the 'conceptual' problems really only serve to benefit kids who have the raw cognitive power not to NEED the practice of grinding rote. That is, anyone can grind procedures and artificially expand their domain-specific working memory (to eventually be able to do the conceptual stuff), but as schools emphasize that less and less, the kids who need it won't do it, and then who's LEFT in the classes are just the kids who are smart enough to grasp the procedures intuitively or w/ little effort. This is ironic of course b/c the liberal educators who want to say that everything is about practice/effort are also the ones who think that 'conceptual' problems are better, when really they reward the kids who were gonna succeed no matter what. But math classes (especially low level math classes like the calc series and linear algebra, which are still often taught computationally, but increasingly less so) that are made conceptual artificially exclude kids that COULD do math if they were made to grind the work, but misguided philosophy of pedagogy stuff prevents them, and then they think can't do math.
Great explanation, love it!
My calc 1 professor did not teach this. We did some log differentiation but I did not know how to do this at first. I am very rusty.
On error-catching routines: one thing I've always tried to emphasise whenever I've taught calculus are strategies by which a student can check their work, even under exam conditions. It is important to go through line by line, but it is also very hard to spot errors that way, since one has a habit of believing oneself. Thus, better are strategies which check the answer somewhat independently. This works especially well for integrals and differential equations, where I always urge students to check their answer by differentiating it (and in the case of a DE, plugging it into the equation). [1]
For differentiation questions this is harder, since of course checking by integrating would be quite hard work! For this, I recommend they identify different techniques for finding the answer (and I try to point out such different routes in examples classes) and to--if they have time--check their answer by solving the question via a different method. Another approach (though perhaps hard to do well under exam stress) is to check, as it were, qualitative properties. Take our f(x) = x^sin x + x^cos x. For x in (pi, 3pi/2) both sin x < 0 and cos x < 0, so f(x) must be decreasing in x. So, if one has a calculator, one can take one's expression for f'(x), compute f'(4), and check that it is < 0. If one has screwed up, this will probably tell you about 50% of the time, and checking that, say, f'(1.5) > 0 will get that up to 75%. This does however require a bit of creativity and a clear head to come up with, so might be a bit much to do in an exam unless one had already finished with a good chunk of time and wanted to really check one's answers carefully.
------------------
[1]: Amusingly, this will also prepare them in the future for more advanced differential equation techniques like finding a weak solution or perturbing the equation, where you might not know if the "solution" you get is a classical solution without plugging it into the DE and seeing if it works.
I think the Q is fair. On the one hand, students need to know the standard concepts and apply the tools. On the other, they should be able to think on their feet and be able to think through unusual situations. A test should have a sensible balance between tools/concepts on the one hand and curve balls on the other. (Say 80:20 - so the 12% can get a fair reward for seeing it but the 88% aren’t too disadvantaged provided they are getting the tools and concepts).
You mentioned time pressure in passing, but I think it deserves more emphasis. Time pressure is exactly what causes students to focus less on their "error checking routines". In my opinion exams put students under this time pressure way too often. Being fast is not what makes one a good mathematician.
Ya I really hate time pressure to, very inauthentic to real math. When I think about the value of procedural fluency, it is actually less for time pressure and more about cognitive load. It can be easier to see through complicated solutions to the big ideas when you are less bogged down by details.
6:52
Is that procedural fluency convered and inculcated in total during the course Work ?
9:03
how much emphasis is given to attentiveness to some details ithe course work ?
It's a bit hard to say. In class we dive into all sorts of details, but the trick is extending that level of attentiveness to when students are working outside of class.
Autopilot is a nice way of putting it. I prefer the expression "The lights are on, but nobody's home".
Very nice and incredible. Thanks a lot from Brazil.
Thank you too!
I think the hard part is that the first step of taking the logarithms hides the linear nature of derivatives. My first thought was to factor it as x^cos(x)(1+x^(sin(x)-cos(x)). Now you can use the product rule, and the results from the product rule don't fully eliminate the problem- you do have the 1 outstanding- but now my pattern brain says 'the derivative of 1 is 0, whats the derivative of the other thing' and I end up with the right answer
I'm proud of myself that I saw that you needed to split it up into two separate derivatives :)
The moment I saw the "hard" problem on the screen I stopped and tried to solve it myself. The first thought I had is that it's a sum, so I just differentiate each term individually and add them at the end.
But the entire time I had the thought in the back of my mind that if this problem is so hard then clearly my solution must be wrong. I guess the teaching I have to take from this video is to trust myself.
Yup! Sometimes the hint of how difficult it is makes one second guess!
I would say math is the poetry of the language rather than the language itself.
When a seemingly-unhard problem turns out to be hard for many students, there is more reason to focus on it in the future. So perhaps it should explicitly reach the chalkboard (I'm showing my age) next year.
I still love the chalkboard!
Oh yes, I usually call it algorithm learning vs conceptual learning. I found out I was algorithmically learning my high school courses and wasn't until calculus where I felt like I was conceptually learning
function of sums is the sum of functions
they're not familiar with that idea, not just the basic linearity of derivatives but that y(x) = g(x) + h(x) where g and h must be solved then put back together. Theres really not much exposure to that idea prior calc and the examples that come come pre calc are very superficial. Did pre-calc ever seriously put interest in quadratic functions being a sum of functions or is just "this is a function look here". Its rarely taught as look, this is the sum of g(x) = ax^2 h(x) = bx and m(x) = c
For you this is the hardest question you have given , but for me it is one of the easiest question i have solved .
I am preparing for Jee mains exam ,please see their previous question paper
Practice procedural fluency, test conceptual reasoning? Sounds to me like this problem would do better as homework than on a test. Imagine if homework was designed to challenge your learning so you actually can think outside the box, instead of being designed as busywork. Yes, it is busywork when you put the same exact problem a dozen times with different numbers, otherwise you're saying the point of the course is to learn the rote tasks of the same handful of easy problems, rather than being to understand how to use a variety of tools to reliably tackle many, distinct, interesting problems.
Ya I think that's a good way to think about it. Generally my tests these days are pretty low on computational frustrations, try to cut to the big ideas as much as possible.
If that's the hardest problem your class must be so easy
Ikr. I can’t believe 12% of students got that right. My god people are retarded
Well it certainly wasn't intended to be a hard problem, but I think my calculus classes are pretty standard in terms of difficulty at least in a Canadian/American first year university context for STEM students.
Hardest is subjective. And based on what Dr. Bazett said in this video, I suspect that hardest, here, is most properly defined as "lowest success rate". I suspect that's different from what you might define as "hardest".
Many students, due to time constraints, have highly developed procedural fluency. That second problem difficulty definitely based on lack of conceptual understanding as far as splitting it into two parts instead of trying to work in mass.
Nice thumbnail with the triple question mark. No choice but to take a peek
12 % of the students were students of math, the rest were students of exam taking.
Before the test is done, a list of exercises should be suggested and, after the test is done, all teachers/professors should solve the test in front of everyone in order for everyone to see what they want in the first place. This is enough for giving regular classes.
The audio might have a desync
I wish I had seen this video when I was a teacher of calculus.
I definitely need that tshirt, where did you buy it?
It's my own merch! Link in description lol!
@@DrTrefor Before writing a comment, I checked in the description and somehow did not see that link. Thanks x)
ive noticed these sort of hangups in my own tutoring of students. ie theyll make mistakes of inertia like this. use a method that they used before on a similar problem for which it isnt appropriate to use that method directly
lmao if this is considered hard in calculus than I'm ready for it.
Sir , i have question in caluclas,
y'+xcosy=x^2
This is a differential equation.
Solve for its solution
Sir please reply your solution
Challenging problems are cool!
I disagree about this being merely a procedural error. Sure, they did not follow the procedure correctly but there's more to it. To many first-year calculus students, the derivative is a black box. When something goes wrong with the process, they don't understand how exactly everything fits together so error correction is much harder. Having some higher level understanding of the derivative as a linear operator or how the logarithm and exponential map between addition and multiplication makes everything feel obvious.
I do think the notion that the derivative is a linear operator is not "seen" by most students, including those whose professors point it out explicitly. They probably know it has this linearity property, but without the breadth of experience with operators in general that this "linear operator" is a salient point might easily be lost and it is just another rule.
If the hardest test problem is finding a derivative then something's wrong
Nice video sir 🎯
From India.
Easy question bcz it's in the example problem of the most basic book for high school maths (ncert maths).
The exercise was not difficult at all: it is a mechanical computation that involves the f(x)^g(x)=exp(g(x)ln(f(x))) trick (for a positive f(x)), and any decent high school student should be able to do it correctly.
Haha 3 years later and now I see why I got that question wrong
I am just not sure why you would want students to do the implicit differentiation here. I mean it works, but the simple logarithm identity there in your second part just applies directly.
New channel BCA Math (Mr high education) 👋
what about when x is negative?
This is an even function so makes no difference
Well of course it's sin(x)*x^(sin(x)-1) + x^sin(x)*ln(x)*cos(x). Just use chain rule
I'm confused... What was so difficult with this problem? It's just a messy solution, but the process is straightforward.
It is very easy to make the mistake that you immediately take the natural logarithm of both sides - mathematicians can make that mistake as well if they aren't very careful, because math problems are confusing and super-precise like that, and this is one reason why a lot of math solutions have shortcuts, like for example the trigonometric series when you solve some partial differential equations, so that you don't always have to do the extremely tedious separation of variables crap every time.
Please. This isn't a hard problem at all. In fact your second method (exp of ln) solves it immediately.
I think procedural fluency problems teach very little. Tho I think their importance is partly giving the students a good practice problem that will be in exams. They'll usually not really know why the procedure worked at first.
I suppose any claim of difficulty is relative to the specific audience. In this case it was hard for this particular set of students.
@@DrTrefor i think this is technique specific, actually. You seem to be a big fan of your first method (calling y the thing to derive). I dislike that method simply because it's slower, tho it's necessary to teach for arc of trig functions for instance - the triangle method is faster for arcs, but students find it slightly hard. It's also strictly necessary in a few other cases - calling it y or multiple other functions of x. It's a more general method tho... Which might be why you like it I'd guess.
I think it's one of those accidents of preferring a technique over another. Tho your preferred first technique would also solve the problem if the student remembered the sum of derivatives is the derivative of the sum... Which they should since it gets pretty hard to derive anything else from that point onward.
As an international high school student, this seems like pretty trivial stuff. Just to be clear, is this high school calculus or college calculus in the United States?
This is pretty standard 1st year calculus stuff in the US. I also agree it is a fairly easy learning objective and that was part of why I was surprised, logarithmic differentiation usually isn't one of the hard learning objectives for 1st year calculus.
@@DrTrefor yeah, that makes a lot of sense. I do have to tell you that i really loved your take on how mathematics courses generally tend to be so overly focused on building procedural fluency over conceptual clarity. that made me think- is that also why a lot of people are turned away from mathematics and more towards other fields like Physics or Economics? the funny thing is that these fields also require a ton of math. The application of that math, however, is what seems to be more appealing. I wonder if the abstractness is an inherent part of pure mathematical rigour and if things like integral calculus, which majorly revolve around trying to manipulate functions into somehow fitting one of the 'procedures' you learnt, can ever be taught in a manner that deviates from pure rigour!
The main mistake is conceptual: the students who failed the question had no clue about the theoretical framework.
The limit is a linear operator. The derivative is a limit. Hence the derivative is a linear operator. Period.
Linearity is such an important property that calculus would be useless without it. Actually that's true in general for all the branches of math and statistics. It's such an important concept that it's scaring to think what would be left of ALL our knowledge if an evil God would prohibit us to use it!
I'm sorry Prof. Bazett but it's you to blame. My advice is to KEEP questions like that and to dedicate an entire lesson dedicated to the goddess Linearityr!
Mindful learning is great... For math students. Everyone else has other things to study and simply can't devote time for mindfulness. In fact universities are often building their courses with the exact opposite philosophy being taught.
I've studied mindfully before. It's great if you want to get 5/10s during your whole university life. It doesn't prepare you for exams, it actually harms you for exams and isn't a great practice if you aren't studying for 12h a day. If you have anything to do at all, that isn't the way to succeed in university level.
The 10/10 dudes will cram, cheat, etc. All the practices the professors say they want to avoid, but end up make it happen.
What university makes you do is simply acquiring past test material from a particular professor or access to the place with the question database - so you can play your odds. Yes, tests require you to develop an ability to make tests. Learning is about the last thing they make happen.
You learn mindfully those things a decade later when you are teaching, which I call a shitty system.
I'm sure you're right about those time pressures, but it definitely sucks. I'd much rather know breadth but understand what I do know very well
@@DrTrefor I actually think tests aren't the best way to cultivate mindful students, to say the least. Unless you are doing math, then it might work.
You have to figure out something else. Unfortunately that is particularly hard in the more basic part of any subject as calculus is.
This is not a simple problem, in my mind. I completely agree that it steps away from calculating and into actually doing some mathematics. Once one has spotted that the log rule doesn't work...One has to see a couple of critical steps . First, that you can break the right hand side down into parts y1 and y2 and treat them as mini problems within the problem. Then secondly solve the y1 problem and use that to complete the y2 problem. On an exam that might be a bit of a leap with the stress of an exam. Kudos to the 12%!
I took Calculus as a student 45 years ago, and I'm not sure I played around much with the x^sin(x) function. It's kind of interesting function to the left of zero having both real and complex solutions.
Dude. I already use it.
The first rule of differentiation is …
But it’s a great problem! Since it removes the autopilot and makes you think.
Oh, just got to that part of the video where you spell this out. :)
What that shirt means? Hippo-tenuse!? 🤨
It's a play on words for how hypotenuse sounds like hippopotamus.
I disagree with the premise here. I would argue that the reason students failed to differentiate the function presented to them is not because we have failed at teaching them procedural fluency. They failed because we focus far too much on procedural fluency, and nearly enough on conceptual reasoning. I think this video itself made an accidental strong case for that point. The only thing that has happened here is that the students have memorized a list of algorithms and rules for how to solve a class of mathematical problems, but the conceptual understanding behind those mathematical problems is missing. Since none of what they have memorized is applicable to task at hand, they simply throw their hands in the air, give up, and remain stuck. Someone who can conceptually reason through the problem, however, will not have that problem at all. This is because solving the problem does not require any algorithm at all. What it does require is understanding it conceptually. This actually goes into my complaints of the education system when it comes to mathematics and the sciences. We teach mathematics horribly, all across the board, and every country in the world seems to have this problem. Even at the elementary school level, we start treating children like computers. We tell them how to add, but not what is adding. We tell them how to manipulate decimal notation, but not what the notation means. In a calculus course, we teach how to compute a derivative, but not what is a derivative. Textbooks will briefly gloss over the limit definition of a derivative, but glossing over what looks like an arbitrary definition and leaving at that is not teaching the students anything. Some people go a little further and try to explain that, geometrically, the derivative is just the slope of the tangent line at a point. Again, this is woefully unhelpful: students are not going to learn anything from that, because many students do not even understand the meaning of the word tangent. Now, by appealing to geometry, you have removed the learning from its algebraic setting altogether. So, you want to teach them that the derivative is the slope of a tangent line, but then, they are expected to compute derivatives symbolically? This is terrible technique. Also, giving a geometric intuition for the derivative that does not apply for the more general setting of vector calculus is actually a pretty bad idea.
Computational/procedural fluency emerges naturally from fostering conceptual reasoning. In other words, a student can learn how to compute derivatives by conceptually learning about what a derivative is/does. But it does not work the other way around. And this is the mistake that education makes, hence why the entire system needs to be overhauled and redone from scratch. We need to start with the very basics of the concepts, guide students through learning about the concepts, and then we can create the algorithms for how to engage in the computation from the understanding of those concepts. Students can learn how to compute derivatives as a natural by-product of understanding what a derivative is and why they should care about it. This makes it more organic, and it avoids the problem of students mindlessly memorizing things without understanding them, and it avoids the problem of students overrelying on algorithms and formulae. We need to teach students not how to follow a procedure, but teach them how to think. Humans are not computers, so we should teach mathematics as if we are dealing with humans, not as if we are dealing with computers.
brilliant
New channel BCA Math (Mr high education) 👋
x^sin(x)=e*(ln(x)sin(x)) power rule , solved
That method also works!
Yeah I could do that in my sleep 🥱🥱🥱🙄🙄🙄
+
i have to disagree on the last part: maths is NOT a language. the notations form a kind of language, but there are real ideas, a complex mathematical world, which is described by the language of mathematics.
if it was just a language, it would not be possible to have synthetic proofs. to be able to do that, you need mathematical objects. they are at the core of maths, not the "language".
The first thing I thought of when viewing this problem was the little exponential trick!
You could rewrite the function as e^(sinx*lnx) + e^(cosx*lnx) and differentiate directly!
It's hilarious just how adding one term complicates the expression, but I'd definitely agree that it's a pattern recognition thing where the students just tend to go on autopilot as soon as their brains recognize the problem.
Edit: Fixed an algebraic mistake, courtesy of Christopher Dudman.
Is that right? e^(sinx*lnx+cosx*lnx) = e^(sinx*lnx) * e^(cosx*lnx) = (x^sinx) * (x^cosx) : but this is not the original function. Apologies if I'm being dense!
@@bizzydizzy You are right IMHO y =x^sinx +x^cosx = (x^sinx)(1 + (x^cos^x)*(x^(-sinx)))
@Christopher Dudman Huh. Yeah, you are correct. Apparently this isn't the correct solution - graphing it against the actual integral shows they're not equivalent...
I used a derivative calculator to see what the solution should be, and it used the exact same method as Dr. Trefor.
Thanks for pointing that out, kudos to you! You've taught me something new!
@@daddy_myers I've been thinking about this overnight. Exponentiate the terms individually, and your trick works a treat- much neater than the method I used. I'll look for opportunities to use this- thanks for the tip!
y = e^(sinx * lnx) + e^(cosx * lnx)
dy/dx = (cosx * lnx + (1/x)sinx) * x^sinx + (-sinx * ln x + (1/x)cosx) * x^cosx
Note to self- must learn LaTeX!
@@bizzydizzy Oh, that's amazing! Thank you for saving me the effort of thinking it through! :)
That's the way I should've written it in the beginning, but I managed to slip it up!
I too wish UA-cam had LaTex, what a shame!
Never forget
The earth is flat
And differentiation is linear
I went straight to the second problem and got it pretty quickly, but it was pretty simple since I wrote the function as e^(sin x log x) + e^(cos x log x) instead of taking logs and doing implicit derivation. Might have caught me off guard too if that was the method I would have been thinking of.
I do not understand the difficulty of this problem honestly, maybe this is related to the american way of teaching calculus. This is why you learn to use: a^x=e^(x*ln(a)) as you mentionned and not the stupid "log-rule" (that may be useful sometimes but should not be the default approach).
Meh, I think "taking logs of both sides" is a pretty common approach in a lot of mathematical contexts. It is true in this case the e^ln^blah trick turns out to be a bit easier in this case but I wouldn't say that is a universally better method. Mostly it is just equivalent.
y = αu + βv, dy = αdu + βdv, …
Skipped to the "hard" problem. Before watching the video to the end, I am just asking if I'm retarded or can I not use the fact that (g(x) + f(x))' = g'(x) + f'(x)???
that's exactly it!
yay i did it right
Hi
hello!
Clickbait. Are you seriously expecting me to believe you struggled/ was surprised with this basic integral? Wasted my time. Unethical, this math not TikTok dances. Respect the intellectual field!
You appear to have run out of ideas for new videos.