an unfair calculus 1 homework question

Рет қаралды 163,694

2 жыл бұрын

I told my calculus 1 students to omit this homework question, i.e. finding the second derivative of x^2/(1+e^x) by using the quotient rule. We can just use the quotient rule to get the first derivative pretty easily but I think the second derivative is very unfair. Not only the process is quite long, but only I can see many of the students will make mistakes when they find the derivative of (1+e^x)^2 because we haven't covered the chain rule for finding the derivative yet.
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Пікірлер: 321

@blackpenredpen 2 жыл бұрын

The chain rule is in sect 3.4 but this problem is in sect 3.2 For more tutorials on the chain rule, see kzfaq.info/get/bejne/gpmlmKyjt9_PaZ8.html

@tabris1135 2 жыл бұрын

actually you do not need the chain rule. e^2x=e^x*e^x, right? product rule :D

@euanmcdougall1917 2 жыл бұрын

@@tabris1135 did you even watch the video lol

@tabris1135 2 жыл бұрын

@@euanmcdougall1917 nah the comment was like 1 minute in.

@apope2087 2 жыл бұрын

Poor course structure if the chain rule is introduced after the quotient rule. If anything; the quotient rule is a (sometimes) quicker method for a combination of product rule and chain rule. (I.e. all problems could be written as U.V^(-1) rather than U/V).

@prithvisukka9271 2 жыл бұрын

@@apope2087 yeah

@Andy-ju8bb 2 жыл бұрын

My maths teacher used to set horrible calc homework. He once had us work out the 4th derivative of a mixed trig and hyperbolic equation with multiple complex fractions, and we were told we could only simplify it at the very end (no simplifying, collecting like terms or clearing complex fractions until the very end). By the time I was done, after about three hours, I had used the quotient rule and chain rule so much the whole thing filled about 10 sides of A4, and then once I finally simplified it, the answer was 1. I cried myself to sleep.

@tenebrae711 2 жыл бұрын

Wow, can you post an initial expression here? I have some free time tomorrow and would like to sharpen my trig and calculus skills

@derekmaggard9235 2 жыл бұрын

Sounds like he just wanted to torture you.

@blackpenredpen 2 жыл бұрын

😮

@Firesawdust 2 жыл бұрын

Don’t you just love when you do all that hard work for such a simple answer 🤣. I’m in Calc 3 right now and you guys would love to see my reactions to homework problems that require so much work but leads to yet a simple answer 😂

@teelo12000 2 жыл бұрын

In the Professional Math Teaching World we call that the "Derivative Rickroll Assignment".

@drpeyam 2 жыл бұрын

My diff eq textbook did the same thing 😡 It assigned a problem about Laplace transforms of shifts in section 6.2 even though that’s in 6.3, so I couldn’t assign that one

@zeychin 2 жыл бұрын

*: Laplace transforms of shifts are left as an exercise to the reader.

@patsplats 2 жыл бұрын

It's called foreshadowing 😅

@SuperYoonHo 2 жыл бұрын

Wow fancy seeing you here dr peyam!

@kummer45 2 жыл бұрын

This is what a great teacher looks like. He's humble, sincere, dedicated and inspiring. His teaching techniques are outstanding. He got extremely better with time. Great work as usual.

@blackpenredpen 2 жыл бұрын

Thank you!!

@Doctor_Drew 2 жыл бұрын

I like how you omitted this problem to not discourage students from long winded algebra. perhaps this could be a bonus or "challenge problem"

@Coni1553 2 жыл бұрын

"I'm not gonna say cool because...I don't really think so 🤷" What a roast

@d_mcg 2 жыл бұрын

If you split the f' fraction, you can factor out a 2x: 2x(1+e^x)/(1+e^x)^2 - (x^2 e^x)/(1+e^x)^2 which reduces to: 2x/(1+e^x) - (x^2 e^x)/(1+e^x)^2 Little bit easier from here.

@moazamabbasi1899 2 жыл бұрын

Love these integral tricks...heheh.

@dqrksun 2 жыл бұрын

I did this too lol

@smith899 2 жыл бұрын

@@dqrksun Me too

@vedantrao2350 2 жыл бұрын

If you don't want to use the chain rule, you can also write the denominator as (1+e^x)(1+e^x) and take the derivative with no problem.

@ultimatedude5686 2 жыл бұрын

This was the obvious solution to me

@stephenbeck7222 2 жыл бұрын

‘No problem’ hehe

@joaopedroalves1200 2 жыл бұрын

And even if we develop this product, it is still possible to calculate its derivative without the chain rule. The e^(2x) term is not a problem, since we can use exponentiation properties and rewrite it as e^(x+x)=e^x*e^x, which is possible to differentiate using only product rule.

@renmamamiya8840 2 жыл бұрын

I'd imagine though it would probably take an extra page to solve it without the chain rule though.

@roderictaylor 2 жыл бұрын

This is why when teaching I like to cover the chain rule before the quotient rule.

@matthudelson3409 2 жыл бұрын

For "entertainment value," work out the general quotient rule for 2nd derivatives (if y = f/g, then y' = (f'g-g'f)/g^2, and y'' = LTR) in terms of f,g,f',g',f'',g'' and then you can plug in the pieces directly.

@TechnoRaabe 2 жыл бұрын

Could you also work out a general quotient rule d(n) for the n-th derivative?

@matthudelson3409 2 жыл бұрын

@@TechnoRaabe It's possible, but unpleasant.

@matthudelson3409 2 жыл бұрын

As a follow-up, there is a nifty formula for your general nth derivative quotient rule that involves determinants. I suspect it's not well-known.

@VikeingBlade 2 жыл бұрын

@@matthudelson3409 Woah, that sounds so cool. You mind sharing it/its name? :)

@theproofessayist8441 2 жыл бұрын

Thank you for being a champ in making the students' lives easier BPRP - you rock!

@blackpenredpen 2 жыл бұрын

😆 thanks

@Anonymationsthecoolanimator 2 жыл бұрын

Even Conan is appreciating BPRP😂

@---kd2sv 2 жыл бұрын

Usual professors be like : I expect you to have previewed the chain rule before I teach chain rule

@blackpenredpen 2 жыл бұрын

😆

@saurabhkujur6331 2 жыл бұрын

Professor, I watched your videos during my depression time and I survived and also learn a lot and I still love watching your videos. So, thank you BlackpenRedpen professor

@blackpenredpen 2 жыл бұрын

Hi Saurabh! How are you doing now? I am happy to hear that my videos have helped you. 😃

@saurabhkujur6331 2 жыл бұрын

@@blackpenredpen now I am good, after watching your videos and lots of talking with my family now I am better and continue to find inspiration to get going forward in my life.

@blackpenredpen 2 жыл бұрын

@@saurabhkujur6331 glad to hear. Take care 😃

@cheifcheepa 2 жыл бұрын

POV: u differentiated this function using the properties of natural log 😎

@dioptre 2 жыл бұрын

how do u do that

@ta_helado 2 жыл бұрын

@@dioptre logarithm differentiation, search it on youtube, lots of viedos

@dioptre 2 жыл бұрын

@@ta_helado ah, ok thanks

@drpeyam 2 жыл бұрын

But you need the Chen Lu to prove that (ln(x))’ = 1/x

@jessebosker122 2 жыл бұрын

@@drpeyam Proving and solving a question are different things 😜

@chessthejameswei 2 жыл бұрын

I wonder if it's better to teach product rule and chain rule and then quotient rule, since quotient rule is a combination of product rule and chain rule. Great video as always!

@blackpenredpen 2 жыл бұрын

Good equation! I would agree! Since when we did the derivative of (1+e^x)^2 by using the product rule, then we can introduce the concept of the chain rule!

@VikeingBlade 2 жыл бұрын

Woah, that's suvh a cool interpretation. I never realized it's a combo of product and chain, even though I usually just say f/g = f*1/g and do it that way since it's easier for me than using the quotient rule

@nimmira 2 жыл бұрын

Back in my day, we would set aside the nominator and denominator and differentiate them on the side (be it homework or exam) and then just copy them. It's like setting the top as f(x) for example and the one below as g(x) and work them separately, then on the answer paper apply the quotient rule or any other rule. I think it helps on keeping the work organized because most of the mistakes that one might do here are the "visual" sort. The person would be good in differentiating but his or her eyes would skip a little parameter and the WHOLE thing gets screwed up. But if such question comes in an exam, the time limit must be considered as well - one of the pains i had to deal with in the past is when the professor makes an exam thinking that students would have the same thought process as him or her and wouldn't realize that they are students and he or she is someone with PhD; The picking time to solve a problem is under-development still for students at this stage. A professor would give an exam of 10 question estimating that each needs 10 minutes or 15 minutes to solve in writing so a typical 90 minutes is just about fair and enough, they would think.

@scienco-gratia 2 жыл бұрын

Good work man Appreciated

@alasdairspitzer4625 2 жыл бұрын

You're WAY TOO KIND...

@Ethan-qq1nf Жыл бұрын

props to you for admitting you were wrong. Love your channel

@nicolastorres147 2 жыл бұрын

9:35 bprp even wrote the terms in the same order 😂😂😂

@blackpenredpen 2 жыл бұрын

😆

@blackpenredpen 2 жыл бұрын

I was surprised when i saw that too 😆

@fanamatakecick97 2 жыл бұрын

This is a perfect bonus question

@waynedayata3775 3 ай бұрын

Now integrate it twice to see if you got the original function

@chih_le 2 жыл бұрын

In Singapore, we teach chain rule before product and quotient rule, followed by applications and only after all that do we introduce trigo, exponential and log derivatives

@george2358 2 жыл бұрын

You don't need the chain rule for the derivative of e^2x, you can write it as (e^2)^x and use the normal derivative rule for exponentials

@hasanalmatrood1441 Жыл бұрын

Or just make it into e^x *e^x and use product rule

@asparkdeity8717 Жыл бұрын

Chain rule is by far simplest, just e^f(x) -> f’(x) e^f(x)

@svxl_ 2 жыл бұрын

Bro I don’t know anything about calculus but this video was very interesting. This is making my brain want to learn more.

@poncho4509 2 жыл бұрын

HELLO!! Always..the good stuff's in the back

@aiaioioi 2 жыл бұрын

our math teacher would do the same thing at the end of homework and when everyone checked the answers in class, there were only 2-3 people who could do these imo, it's still important to do, so the students are not afraid of complicated things

@robertcotton8481 2 жыл бұрын

Reassign later when learn chain rule makes good example of how can be done faster

@GodzillaFreak 2 жыл бұрын

You still have to do quotient rule and product rule, which are the really rough parts

@jackson9143678 2 жыл бұрын

@0:32 That face is so priceless, love it :)

@miikey_lol 2 жыл бұрын

Can also be done using Leibnitz' Theorum letting f(x)=x^2 and g(x)=(1+e^x)^-1 Then the derivative is f''(x)g(x) +2f'(x)g'(x) + f(x)g''(x)

@sr.tarsaimsingh9294 2 жыл бұрын

01:12 Finally Sir today I find yours name.

@xinpingdonohoe3978 2 жыл бұрын

Only now?

@sr.tarsaimsingh9294 2 жыл бұрын

@@xinpingdonohoe3978 yupp

@richardspencil 2 жыл бұрын

This is when I would use logarithmic differentiation.

@Risu0chan 2 жыл бұрын

Sometimes it's easier to use the product rule than the quotient rule, especially with a denominator to some power; that is, instead of blindly using this formula (f/g)' = (f'g - fg')/g² , where g is already a square (or a power) use that one (f × 1/h²)' = f' × 1/h² + f × (-2h'/h³) It took me about 30s. Add 30s to regroup over a unique denominator if needed/asked.

@jeremydavis3631 2 жыл бұрын

That's true, and very useful, but what you did here uses the chain rule to get from 1/h² to -2h'/h³. He set out to do it without the chain rule.

@mathe.dominio4765 2 жыл бұрын

💙🙏

@user-nj1wk3ez7p 2 жыл бұрын

actually it's not necessary to use the chain rule to figure out the derivative of e^(2x). We can just rewrite it as a product e^x * e^x ;)

@tobybartels8426 2 жыл бұрын

I came here to say this! (Actually, I was going to say that you can apply the quotient rule twice and divide by both factors of 1+e^x separately, but your way is better.)

@AsilKhalifa 2 жыл бұрын

ah that's very smart I didn't think of that!

@TomTom-se4of 2 жыл бұрын

We can also rewrite it as (e^2)^x and use the formula for the derivative of b^x

@tobybartels8426 2 жыл бұрын

@@TomTom-se4of Good idea! Some books cover e^x before the Chain Rule but save the more general b^x until after the Chain Rule (because you can derive it with the help of the Chain Rule). But a textbook certainly could cover b^x in general before the Chain Rule, so if this book is like that, then your idea would work.

@VikeingBlade 2 жыл бұрын

@@TomTom-se4of Oh wow, you're so right! That's so cool :)

@okok7148 2 жыл бұрын

we had a test with a function like that two weeks ago. nobody ended it and the teacher blamed us for being too slow.

@kent631420 2 жыл бұрын

0:36 I have never ever seen mr.steve swear before XDD That should *not* be in the product/quotient rule section

@Dreamprism 2 жыл бұрын

0:34 I burst out laughing as you came to your realization

@technoviking Жыл бұрын

Numerator and denominator are both > 0 for x0, so take logs of both sides first, then differentiate.

@TheSpoonThatDied 27 күн бұрын

This is that question that isn't even too hard, it's just so stupid that it makes you wanna rip your hair out

@wildmonkiesJR 2 жыл бұрын

a real leader... leading by example

@anshumanagrawal346 2 жыл бұрын

Beyond a certain point, it's just mindless algebra, and that doesn't really help

@robbychu6726 2 жыл бұрын

After that, check if the answer is correct by integration

2 жыл бұрын

As a non-native English speaker living in Hungary, I always heard "combine like terms" as "combine LIGHT terms" and I always wondered why that usual process was called that way in English ;)

@mathe.dominio4765 2 жыл бұрын

💙🙏

@volodymyrgandzhuk361 2 жыл бұрын

@@mathe.dominio4765 ¡hola!

@jumpman8282 2 жыл бұрын

Reminds me of when I heard "x of 2" instead of "x sub 2" (x₂)😄

@polygonc4538 2 жыл бұрын

Haha funny, I also use "Thomas Calculus" for my Calculus exercise classes ;)

@blackpenredpen 2 жыл бұрын

This is actually from Stewart.

@polygonc4538 2 жыл бұрын

Oh ok, I see... For me it had very similar type of layout

@pauljamesmontoya4654 2 жыл бұрын

currently taking calc 3 and let me tell you i DO NOT miss these calc 1 nasty problems. good luck for calc 2 integrals tho i hate them especially surface area and trig sub.

@pauljamesmontoya4654 2 жыл бұрын

area of surface revolution ^^

@luxo1035 2 жыл бұрын

Hey BPRP, here's a very small fun challenge, if you want to do it: Find the indefinite integral of sqrt((-x^2 - x + 1)/(1 - x^2)) with respect to x.

@chrisglosser7318 2 жыл бұрын

It’s a lot easier if you cancel the exponential in the numerator before you calculate f”

@deep24543542 2 жыл бұрын

I learned in calculus I could simply write smaller on blank paper to fill less space and hope I don't have to go back and try to find a mistake

@VikeingBlade 2 жыл бұрын

One of my friends writes math so small. His whole homework set will be half a page while mine is like 14 pages

@MonzennCarloMallari 2 жыл бұрын

It's not too bad. Lots of algebra, which means lots of places to make careless mistakes, but that's why this is homework for that section and not a time-limited test question for that section.

@science-explained_ka 2 жыл бұрын

I've been enjoying your classes. You're good at giving explanation. I tried to join your membersh using my debit card but not successful. What should I do?

@k_wl Жыл бұрын

u can just square the bottom fine and for d/dx (e^(2x)) u can write it as d/dx(e^x * e^x) and differentiating it gives 2*e^(2x), just a thought :)

@TheWizard856 2 жыл бұрын

Those frivolous page long problems lol

@vincentstone7272 2 жыл бұрын

Rewrite e^2x as e^x * e^x. Use product rule

@robertveith6383 2 жыл бұрын

No, that would be e^(2x) = e^x*e^x.

@vincentstone7272 2 жыл бұрын

@@robertveith6383 is that not what I said?

@mreverything7056 2 жыл бұрын

Everybody gangsta till the math teacher starts cursing 0-0 But realistically, I really respect how open and realistic you are, even about little mistakes you make. Really cool!

@krishgarg2806 2 жыл бұрын

I like how this is a type of question our physics teacher would give, after he taught us the very basics of differentiation and integration to be used in physics. Actual calculus in maths will be taught in next academic year lol. But the fun thing is, even with those basics I solved this.

@JustBackgroundNoise 2 жыл бұрын

The textbook authors probably put that in there so they can change it and justify making another edition.

@davidst.arnauld6763 2 жыл бұрын

What a bummer!! BTW, I learned the chain rule first.

@poosywoosy5553 2 жыл бұрын

As a somewhat decent math student when I was younger, now I don’t understand math when math is only good in measurements of size, money, or count. Else every time I use math in life, it feels like value changing for a greater purpose of stuff I can’t really fathom.

@tobybartels8426 2 жыл бұрын

You can't really do this here when you're avoiding the Chain Rule, but normally if you're dividing by a power (or root), you shouldn't use the Quotient Rule. Instead, rewrite it as multiplication by a power with a negative exponent; you're going to have to use the (generalized) Power Rule anyway, and it's hardly worse with a negative exponent, but now you get to use the Product Rule instead of the Quotient Rule. (If you _really_ hate the Quotient Rule, you can always get rid of it by using multiplication by a power with an exponent of -1, but now you've made more work for yourself because you have to deal with the power. So I only avoid the Quotient Rule this way I've already got a power in the denominator.)

@kyoyoberry 2 жыл бұрын

If I may ask, what is the Question Rule?

@tobybartels8426 2 жыл бұрын

@@kyoyoberry It's the one where you try to type ‘Quotient' and your phone thinks that you meant to type ‘Question’. 😭 ETA: Fixed now.

@giorgostarnaras5658 2 жыл бұрын

you can also take the limit definition to find the derivative of (e^x+1)^2

@pelledanasten1615 2 жыл бұрын

Yes, but you need to know d/dh (e^2h), so you'll still be using either the product rule or chain rule.

@blackpenredpen 2 жыл бұрын

Even better 😆

@robertveith6383 2 жыл бұрын

@@pelledanasten1615 -- You mean of e^(2h).

@pelledanasten1615 2 жыл бұрын

@@robertveith6383 yes

@abdirazksomo88 2 жыл бұрын

i was realy understand but i was confused the appliction you checked were we can gat that applictin or share us it's link coz some times we need as we are student level

@GhaliumHD 2 жыл бұрын

I think i recognized that book lmao, thats from early transcendentals by anton 11th edition?

@blackpenredpen 2 жыл бұрын

Stewart.

@zinave 2 жыл бұрын

It's pretty cool to understand calculas as a 16 year old

@mathe.dominio4765 2 жыл бұрын

💙🙏

@user-fw9ld7jx8r 2 жыл бұрын

Thanks 😊

@kingbeauregard 2 жыл бұрын

Use Weierstrass substitution, DUH!

@Sovralin 2 жыл бұрын

Where can I join your classes?

@joekerr3638 11 ай бұрын

My calculus teacher said the hardest part of calculus is the algebra; this problem exemplifies that maxim.

@fromblonmenchaves6161 2 жыл бұрын

Irrelevant but, for extracting values, which one is better for precision? rounding or truncating?

@divyanshsharma6791 2 жыл бұрын

Now please integrate the function to check the answer

@dennishou5944 2 жыл бұрын

It's not that bad I've had physics problem sets where taking a derivative ends up taking a whole page :(

@n0mad385 2 жыл бұрын

it must be amazing have a famous youtube as a teacher

@vinceesperanza250 2 жыл бұрын

Sir what calc book are your students using?? thanks..

@virajfunhouse8491 2 жыл бұрын

Road to 1 mil

@kquat7899 2 жыл бұрын

Ridiculous question!

@DatBoi_TheGudBIAS 2 жыл бұрын

If blaclpenredpen swears it's real shit here 0:32

@jensraab2902 3 ай бұрын

I went a slightly different way to get around the chain rule. Rather than writing (1+e^x)(1+e^x) I expanded (1+e^x)² into (1+2e^x+e^2x) as usual but simply wrote e^2x as e^x * e^x. Essentially the same idea just writing a square as two factors at a different spot.

@Pseudoku_RL 2 жыл бұрын

I'm really glad I didn't have to do too many problems like this. Understanding calculus, to me, garners it's difficulty from the concepts, not the application. Making the application needlessly difficult always seemed like a brutal way of punishing the students and making them not enjoy the course or the real beauty of calculus. To me, calculus should be an engaging puzzle, not a mental hellscape lol

@OOO-wv4qk 2 жыл бұрын

你拯救了我跟室友

@prasadtutor2732 2 жыл бұрын

What I’ve we cross multiplied to y(1+e^x) = x² ? Now you can find the second derivative easily from this and use the result for y’ ..

@VikeingBlade 2 жыл бұрын

Yes!! Someone was just talking about that. That works so well. Great idea :)

@SISKCERTWaJaVlogs 2 жыл бұрын

Chad challenge: Get the double deriv using the difference quotient

@VikeingBlade 2 жыл бұрын

This is one challenge I simultaneously want to do and don't

@KazACWizard 2 жыл бұрын

lets be honest, if i had you as a teacher, you could literally tell me to avoid the question, but all it will ever do is make me wanna do it more. thats the point right? its so annoying or hard that me getting it right is actually going to feel like a little achievement.

@VikeingBlade 2 жыл бұрын

Once we were doing parallelograms in geometry class. We learned them on the Euclidean plane and moved on to coordinate geometry. Then we got to a coordinate geometry problem that had parallelograms, and he wanted us to use the sides only, not the angles. I wondered if I could find angle measure on a coordinate plane. I went to my teacher and asked. My teacher told me not to worry about doing that. I went home and looked up 'angles on coordinate plane.' That led me to the law of cosines. That led me to sohcahtoa. That led me to Algebra II. That led me to calculus. That led me to petition my school to take calculus two years early. That led me to graduating college with a bachelor's in math five semesters early. Never underestimate the power of a boy/girl's curiosity

@mikeredd2608 2 жыл бұрын

now integrate that

@maxime3118 2 жыл бұрын

How old are your students and which country are you from so i can compare to my country level ?

@jeffstryker2419 2 жыл бұрын

This is fine for a homework question, they are supposed to be little harder since students have longer to think and work. It might be too long for an exam question.

@General12th 2 жыл бұрын

Just so long as we don't expect students to get a perfect score on every homework assignment, then yes. Problems like this are stretch goals.

@stephenbeck7222 2 жыл бұрын

I guess, but I still don’t want students taking 15 minutes on a single homework problem that’s just a humongous calculation.

@blackpenredpen 2 жыл бұрын

Most students would miss the “chain rule part” bc we don’t have the experience yet. Either they will get 2(1+e^x) or 2e^x+e^(2x) for the derivative of (1+e^x)^2 or 1+2e^x+e^(2x). It’s not clear why we would need to write it as (1+e^x)(1+e^x) then differentiate. It’s simply not a fair question to be asked in this section.

@duckymomo7935 2 жыл бұрын

No, no one computes derivatives like these by hand anymore This job is relegated to wolfram alpha

@bobh6728 2 жыл бұрын

@@duckymomo7935 do you want to be the person that uses Wolfram Alpha, or the person that programs Wolfram Alpha?

@jglf 2 жыл бұрын

Now integrate it to check :p

@ExpertMathematician 2 жыл бұрын

Maybe you can seperate it first to u/v such as 2x/(1+e^x)^2 + 2xe^x/(1+e^x)^2 - …

@ryantsang7144 2 жыл бұрын

I would say you are surely a good teacher at all because you will ask the students not to do sth meanwhile other teachers will say " fk it they can do it " even the questions are non sense haha

@krishgarg2806 2 жыл бұрын

Now you find the 3rd derivative.

@ItzMezzo 2 жыл бұрын

For the next video find f'''(x) x )

@ShapelessMonstrosity 2 жыл бұрын

Learning the quotient rule before the chain rule seems strange to me (I'm fairly sure the chain rule was the first of the 3 that I learned), but I guess there's no reason to teach either the chain rule or product rule before the other.

@mathe.dominio4765 2 жыл бұрын

💙🙏

@dtnicholls1 2 жыл бұрын

I vote you reassign the question... But only because I'm not your student!

@wckewhbckea 2 жыл бұрын

now derive that

@iyanically 2 жыл бұрын

now do the third derivative. :)

@simonwillover4175 2 жыл бұрын

4:42 Were gonna factor out 1+e^x because ... I'm not going to say that's cool (it's just not) Oh you're so amusing! I totally felt the same way wayching you do all of this. However, I think factoring takes up less space is computationally easier for computers to handle. Also, factored results, strange as they may seem, often help with other advanced maths.... so it's useful, but not exactly cool.

@jkid1134 2 жыл бұрын

Your students are going to be soft by the time they get to homework problems which take an hour two semesters later

@blackpenredpen 2 жыл бұрын

Not all my students are studying STEM in the future. For the ones who are, I will tell them to do 100 derivatives in one take 😆

@blurryfca3 2 жыл бұрын

Chain rule should be taught before the product and quotient rules imo

@chaiotic 2 жыл бұрын

Although it's not as elegant, you can also do (1 + e^x)^2 -> 1 + 2e^x + (e^x)^2 -> 1 + 2e^x + e^x * e^x, so the derivative is: 0 + 2e^x + e^x * e^x + e^x * e^x using product rule on e^x * e^x

@awsomn3sss36 2 жыл бұрын

I feel like you could take the ln of both sides and simplify it then take the derivative, but I don’t know if it works and am too lazy to try.

@volodymyrgandzhuk361 2 жыл бұрын

How do you take the logarithm of a sum?

@awsomn3sss36 2 жыл бұрын

@@volodymyrgandzhuk361 it would just spectate (on the right) into ln(x^2)-ln(1+e^x) so you still end up using the chain rule when you take the derivative, but you avoid using a really annoying quotient rule.

@volodymyrgandzhuk361 2 жыл бұрын

@@awsomn3sss36 his students had NOT learnt the chain rule yet.