Calculus Teacher vs. Power Rule Student

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Definition of derivative vs power rule! Want to learn more about calculus with visual and physical intuitions? Head to Brilliant brilliant.org/... (20% off with this link)
The definition of derivative is the foundation of differential calculus. However, many students tend to ignore it and want to use the "shortcuts" (differentiation rules such as the power rule). Not only do we need the definition of derivative whenever we need to determine the derivative of a function, but I also give a calculus 1 problem where using the definition of derivative is easier than using the differentiation rules!
Check out "when d/dx(x^2) becomes the hardest question on your calculus final": 👉 • calc 1 final be like (...
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29 сен 2024

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Комментарии : 442

@blackpenredpen 2 года назад

Check out "when d/dx(x^2) becomes the hardest question on your calculus final": 👉 ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-wUD7n_ohMQk.html

@masternobody1896 2 года назад

my brain stop working i dont get it

@protoxin_ 2 года назад

Ur fan sir

@andrewgregory8859 2 года назад

“The pen is blue” ~ Jim carry

@space_engineer17 3 года назад

Math teacher: spends 1 hour explaining a method Math teacher: and,there is an easy method too

@blackpenredpen 3 года назад

😂 😂 😂

@nei2870 3 года назад

Lmaoo it do be like that doe

@user-cq9hb1pg3u 3 года назад

💗

@shadmanhasan4205 3 года назад

Easy method's only good depending on the situation and function. Useless in Proofs, but convenient for applied calculations.

@batcout8787 2 года назад

And then there's the feeling of proudness once you master the hard method

@earl8295 3 года назад

5:55 I laughed at that facial expression, "can we use L'hopital's rule?"

@OverlordOfDarkness95 2 года назад

Funnily enough, when I first encountered the formal definition of a derivative in a real analysis textbook at university, my first question was why I was required to spend so much time in high school memorising the rules when just working from the definition was intuitively so much easier. Hindsight is often very ironic like that.

@SmallSpoonBrigade 2 года назад

This right here is why high school calculus shouldn't count for college credit. Sure, you're technically doing calculus, but I could probably get my 12 year old nephew to do that much. Part of the point of the classes is that you've got at least some understanding of why the results are what they are, so if something goes wrong, you have some hope of knowing and fixing it.

@tabithal2977 Год назад

@Autistic Doing and Thinking idk that seems like it only happened in your highschool calculus class. In my ap calculus class we learned all the definitions and stuff, did the derivatives once and then just memorized our results so we didn't have to redo the work again.

@chloemccarthy2297 3 года назад

Honestly Calculus is what made me want to be a mathmetician. I cannot explain why, but integrals were so seemingly magical to me as well as differentiation. Hope I'm smart enough to get my PhD in applied mathematics after my Bachelors!

@drpeyam 3 года назад

Whoa that is amazing!!!!!

@blackpenredpen 3 года назад

Impressive, I know that too, you know?!

@martinespinosa3396 3 года назад

WHOAAA

@JasonOvalles 3 года назад

Student wants to use the power rule? "Oh that's brilliant! How did you come up with that?" If they don't know how it's derived - "that's why we're practicing using the definition of the derivative, so we can use the definition to prove interesting patterns we might notice." If they do know how it's derived - "that's great that you've already learned some of this! I don't want to jump ahead so we can give other students time to think and process. But maybe consider yourself a resource for other students in this first unit!"

@jessehammer123 3 года назад

Absolutely, that’s a great method of probing your students’ advancedness without being judgmental toward either them or the other students.

@matthewcrawford4316 2 года назад

I'm big-time guilty of this lol I read the text the night before the lecture so I'm usually the most vocal in my calc II class. I feel bad at times because I'm trying to push myself further in math and I'll ask questions that are a bit more complicated than the rest of the class is at, but at the same time I have people asking how to differentiate 3x (an actual question we had about 4 weeks ago, do not ask me how they made it to calc II) and I don't feel as bad

@MonsterIsABlock 3 года назад

Inaccurate, calculus teachers actually just throw a bunch of formulas at you, bprp is just built different.

@skylardeslypere9909 3 года назад

It seems like you just had some had calculus teachers

@好吧-h6k 3 года назад

both of my parents who were professors in china hate when I use derivative rules for a basic function

@angelmendez-rivera351 3 года назад

@@skylardeslypere9909 It appears most calculus teachers are bad calculus teachers. Tee hee. But in all seriousness, the reason most calculus teachers are bad is because the recommended curriculum for calculus courses is itself garbage. Calculus is taught way too much like they are teaching how to do real analysis in every wrong way possible to get the right answer, rather than actually teaching how to it the right way.

@Ranoiaetep 2 года назад

@@好吧-h6k You probably shouldn't be using the derivative rule for basic functions, just like bprp mentioned he would also just use the power rule if he was randomly asked on the street. But you should be familiar with the mechanism behind it, so you are not stuck when you see a different function that haven't learnt a formula for yet.

@SmallSpoonBrigade 2 года назад

Interesting, neither of the Calculus teachers I had did that. Later on, they would, but we already knew the definition of the derivative and could generally do that on our own if we needed to. It got a bit more annoying at times with transcendental functions, but it could generally still be done with the use of the proper identities.

@tsawy6 3 года назад

Man, that limit of 2^h-1/h is pretty crazy man. Pretty triv w/ L'hopitals, but seeing that you used it in your search for derivs in your other video (and abusing the fact that you can get derivs of all the exponentials from just e^x) I tried to explicitly prove the case for e. Surprisingly tricky, and naturally one of those hard to search sorta math problems. Eventually managed it using epsilon delta and the logarithm definition of the exponential (ln(x+1) as integral from 0 to x of 1/t+1, actually the first rigorous def I was given in uni!) and the taylor expansion around 1+x of 1/x [which is gettable naively]). This was really helpful in taking care of the extra 1 in the limit numerator. Pretty cool problem!

@biffwebster1212 3 года назад

It's 2:11 AM. I haven't taken a math class in roughly 15 years. I've never taken a calculus class. I have no idea what you're talking about. I have no idea why RU-vid brought me here. With all this being said for whatever reason...I LOVE IT. I guess the tube knows me better than I know myself.

@fennecbesixdouze1794 2 года назад

Linear approximation is even easier: The linear approximation of 2^x about x=0 is 1+x*ln(2). Let's set a = ln(2). So we need to find the linear approximation of ax(ax-1)(ax-2)(ax-3)...(ax-9). By the binomial theorem, or even by simple inspection, the linear term here is -9!a*x. The constant term is 0. So we have a linear approximation of -9!a*x, the slope of which we can read off as -9!*ln(2).

@noormann4376 Год назад

We can do the above problem by direct differentiation Simply take natural log of both sides then differentiate normally by chain rule and proceed to find the answer. You might run into some problems with 2^x-1 but u can cancel it out with first term of the fractional part and all other terms proceed to become 0

@forklift1712 3 года назад

I like how you put -(9!) instead of -9! for clarity.

@janeza382 2 года назад

Take this *9! ln1/2*

@k_wl 4 месяца назад

you could just do the question like this f(x) = the given thing we can just assume 2 new functions g(x) = 2^x-1 h(x) = (2^x-2)(2^x-3).....(2^x-10) so in this case f(x) becomes f(x) = g(x)*h(x) and here we can use product rule :D f'(x) = g'(x)h(x) + g(x)h'(x) and at x = 0 f'(x) = g'(0)h(0) + g(0)h'(0) here actually, we see that g(0) is just 0, so it doesnt matter what h'(0) is cuz the final result would just be 0 we know that --> g'(x) = 2^x ln2 from differentiation rules, so g'(0) = ln2 and h(0) = (-1)(-2)(-3).....(-9) = - (9!) so f'(0) = (ln2)(-1)(9!) + (0)(h'(0)) f'(0) = - 9! ln2 it isnt gonna be faster than definition but with enough practice its faster, in my case i didi t like this way faster in my head

@BryanHsieh-c2q 3 года назад

I'm a highschooler currently learning calculus, this is amazing. Let me realize that the ahortcuts were not always available. Such an interesting concept.

@SmallSpoonBrigade 2 года назад

If you've got time, find an online copy of a very old Calculus book from a hundred years or more back. There's a ton of things that are now standard that at the time were rather obscure or completely unproven that we can now use without having to prove them out every time.

@RedditMeIn 2 года назад

in my country a^x where a>0 the derivative has a rule, (a^x) ' = a^x • lna that's why e^x's derivative is the same cause the lne is equal to 1

@Wolf-yr1qy 2 года назад

Somehow I can feel his anger through the screen. As a student, I always loved the limit definition (It was reasonable and easy to understand). I remember when I tried to ask my teacher if I could use syntheic division instead of polynomial division, and she got reasonably angry with me because it was just a shortcut without knowing how it truly worked.

@Firetech2004 2 года назад

3:06 no it’s not. To find derivative of 2^x we use a^x rule which is a^x ln a. So the answer is 2^x ln 2

@anshumanagrawal346 3 года назад

I'm stealing that example

@blackpenredpen 3 года назад

Be my guest! 😃

@Doitcreative 3 года назад

I am start to learning differential calculus in our school but your teaching and sums are really interesting thanks🙏

@BSENKevin 2 года назад

I like the fact he kept the pokeball in his hand the whole time. I dunno why, but that kept my attention just as well as the math.

@caleblechler8929 Год назад

I like how he said “ if I had to do this on the street”

@ninja_trend.. 3 года назад

Hello sir, I am from India and I love the way that how easily you solve these questions in a funny way❣❣

@averageaf4321 2 года назад

d/dx of 2^x is (ln2)e^(xln2), which shares only one point in common with x(2^(x-1)) when both are plotted. So using the power rule for 2^x doesn't work.

@icew0lf98 2 года назад

well then the easiness relies on the knowledge of lim(2^h-1)/h, maybe bruteforcing it is faster than figuring out how to calculate that limit

@Abhishek_Deshmukh8788 2 года назад

"My Hardwork"👍😁

@murahat98 2 года назад

Please make a tutorial on how to switch markers like you :3

@JDC2890 3 года назад

y = x^n lny = ln(x^n) = nlnx (1/y)dy/dx = n/x dy/dx = ny/x = nx^n/x = nx^(n-1) And that's how I'd prove the power rule for differentiation. Not quite sure how i'd prove implicit differentiation works, or that d(lnx)/dx = 1/x.

@DutchMathematician 2 года назад

@YDC2890 Actually, you're not using implicit differentiation here, just the chain rule. You would still have to prove that d(ln(x))/dx = 1/x of course... If you assume d/dx(e^x) = e^x as known, this is not that difficult. Take the following function: y = e^ln(x) Differentiate both sides with respect to x, using the chain rule: y' = e^ln(x) * (ln(x))' Since y = x (by definition of y), this should equal d/dx(x) which is the constant function 1. Hence: 1 = e^ln(x) * (ln(x))' Therefore: (ln(x))' = 1 / e^ln(x) and since e^ln(x) = x we get: (ln(x))' = 1 / x

@pinguyt9496 3 года назад

RU-vid algorithm has ascended to the next level. This is exactly what we're doing in school rn

@filipecosta2181 Год назад

What is the difference of writing (f(X+h) - f(X))/h and (f(X) - f(Xo))/X-Xo ?

@sofyanox12 3 года назад

what an amazing example!

@BHFJohnny 2 года назад

The main question I stared asking and also main reason I quit school is: How does this help solving climate criris, the most serious problem of our age.

@Dhukino 3 года назад

question: Isn't f'(0) the derivative evaluated at zero (so apply def of derivative first, then plug in 0)? What allows us to plug in zero during the process of applying the defof the derivative? some continuity reason?

@angelmendez-rivera351 3 года назад

f'(x) = lim [f(x + h) - f(x)]/h (h -> 0) is true for every x for which the latter exists. This necessarily implies f'(0) = lim [f(0 + h) - f(0)]/h (h -> 0). Your comment seems to suggest that "plugging in" is an operation in itself, but this is not the case.

@ieatgarbage8771 2 года назад

Finding the derivative is much easier when you already know the derivative

@oki3873 3 года назад

Gandalf with pokeball

@rayniac211 2 года назад

Beautiful!

@deltax7159 3 года назад

lol on the street

@TrueSpeak-TS 2 года назад

The students are also fans!!!

@reyadhaloraibi3387 2 года назад

Amazing!

@TheSafeSword 2 года назад

thank u bprp

@sigurdolsen3653 3 года назад

And he is still holding a ball…

@aMartianSpy 3 года назад

prof bprp needs a reboot

@rajashreerao448 3 года назад

man u looked good without that beard and with hair

@calculuslover2078 2 года назад

Well formulas for derivatives are good, but you need to know what the derivative is, and how these formulas are proved.

@prakharjain21 3 года назад

How was using the definition faster. Using the product rule, we can get the answer in a single step. Maybe not the correct example to demonstrate.

@angelmendez-rivera351 3 года назад

The product rule does not give you the answer in a single step, since the product has more than two factors, and it takes various steps to get the derivative of each factor using derivative rules.

@prakharjain21 3 года назад

@@angelmendez-rivera351 yea, but all but one factors will give 0 due to the (2^x - 1) factor.

@angelmendez-rivera351 3 года назад

@@prakharjain21 They only give you 0 _after_ plugging in, which you can only do _after_ finding the derivative via the rules. If you plug in first and the differentiate, you get 0 as the answer to the exercise, which is wrong. So, you are still wrong: the product rule just takes more steps.

@prakharjain21 3 года назад

@@angelmendez-rivera351 hmm, yea product rule takes more steps if you need to write it down. i have a habit of doing all those steps in my mind that's why found it to be short

@angelmendez-rivera351 3 года назад

@@prakharjain21 Well, the definition of the derivative can also be done in your head, though, and it takes like 0.5 seconds of thinking. Writing makes _everything_ longer, it doesn't make one method longer and another method shorter. That's not how writing works, and it isn't physically possible for it to work that way.

@mukhtaarjaamac8763 2 года назад

You know maths but no the way to convey first start aljebra rise over run and then go on

@mikevids8107 2 года назад

I felt so smart in calc when I figured out the power rule on my own while doing the homework in my class but then the next day my teacher had a lesson on it and everyone else learned it too.

@blackpenredpen 2 года назад

That’s great you figured it out on your own. I remember some guy in my AP calc class yelled out the power rule before I could even try to figure it out.

@arsh9908 2 года назад

Gatekeeping formulae? Jk😂

@toirmusic 2 года назад

SAME

@professorpoke 2 года назад

It hurts man. I can feel. 💔

@dyhsehehb6232 2 года назад

Yeah similar kind of thing happened in my class too In 9th class i knew structure of dna and other stuff related to that but then in 1 class our teacher taught everyone that, it hurted man Because before that only i knew and i felt smart before that(btw structure of dna is basically 11th class topic...)

@haakon52693 3 года назад

Impressive! I know this too, you know

@blackpenredpen 3 года назад

😂

@yashmangal3663 3 года назад

@@blackpenredpen nice beard😉

@wenkoibital4779 3 года назад

@@blackpenredpen I am here to serve beard cut for those who do not have beard cut themselves.

@colleen9493 3 года назад

I don’t get the joke can someone explain?

@blackpenredpen 3 года назад

@@colleen9493 watch from 2:10 for like 20 seconds.

@monke4216 3 года назад

The fact that he just marked that timestamp as "final words"!!😆

@johnchessant3012 3 года назад

You can solve the (2^x-1)...(2^x-10) with the product rule too, namely (fgh)' = (f')gh + f(g')h + fg(h'), etc. In this case all terms except the first one vanish and the first term is -(9!)log(2).

@yoav613 3 года назад

The students always win!!😃

@liamwelsh5565 3 года назад

Can you prove this with the limit defination though? That is the whole point of the limit definition is to prove these "shortcuts".

@johnchessant3012 3 года назад

@@liamwelsh5565 I'd prove the multiple product rule by induction from the usual product rule. I guess you could prove it directly with the limit definition but instead of adding and subtracting just f(x+h)g(x) you'd need an extra pair for each extra function if that makes sense

@MH-wz1rb 3 года назад

Would be shorter in a sense to just make the first term f and the other 9 terms together to be g I suppose

@PontusLarsson1 2 года назад

If you did this, you would get zero points, as only your f is differentiable at x=0. The rest of them are NOT differentiable at x=0! I.e. you cannot apply the product rule at all!

@nicolastorres147 3 года назад

Q: Is it possible to have a question so that it is actually easier to solve with the definition of derivative rather than the differentiation rules, which limit is actually easier to solve using the epsilon-delta definition rather than the limit rules?

@skylardeslypere9909 3 года назад

The problem is that to use ε-δ you already have to know the limit. Which begs the question, how did you find that limit?

@nicolastorres147 3 года назад

@@skylardeslypere9909 Maybe you have intuition about the limit (which allows use of epsilon-delta) but don’t know how to actually calculate it using the limit rules? 🤔

@skylardeslypere9909 3 года назад

@@nicolastorres147 Or, I guess, you could use numerical methods to find the limit (well, guess the limit). I'd highly suspect that in the case you suggested, the limit would have to be zero or something simple

@ketofitforlife2917 3 года назад

Linear approximation?

@angelmendez-rivera351 3 года назад

@@skylardeslypere9909 *The problem is that to use ε-δ you already have to know the limit.* You absolutely do not have to know the limit to find it using the ε-δ method. For example, consider f : (-t, t) -> R with f(x) == x, and let x -> 0. There exists a real L such that for every real ε > 0, there exists a real δ > 0 such that for every real x in (-t, t), 0 < |x| < δ implies |x - L| < ε. |x| = |(x - L) + L| >= ||x - L| - |L||, which means that -δ < |x - L| - |L| < 0 or 0 < |x - L| - |L| < δ, implying |L| - δ < |x - L| < |L| or |L| < |x - L| < |L| + δ. |x - L| < |L| or |x - L| > |L|, meaning |x - L| is not |L|. This means either x is not 0 or x is not 2·L, or else L = 0. But the former is impossible since 0 < |x| < δ implying |x - L| < ε must hold true for x = 0 since 0 is an element of (-t, t). Therefore, L = 0, if L exists. Of course, this was tedious, but it was still possible to find L without knowing L a priori, using only properties of absolute value inequalities.

@mr_meow_77 3 года назад

What is inside ur Pokemon ball 🤔 ?

@williamcarver7590 3 года назад

His mic.

@TheCodeSatan 3 года назад

Nice video! Try f(x) = 0 if x = 0 and f(x) = x²sin(1/x) for x not equal to 0 The derivative for this function isn't continuous and hence normal differentiating techniques don't work...we have to go by the definition of the derivative

@DutchMathematician 2 года назад

@ridayesh parab Another famous example where we HAVE to use the definition of the derivative is the following function: f(x) = 0 if x = 0, else f(x) = e^(-1/x^2) If you have a graphing calculator, take a look at it's graph near x = 0. Zoom in, zoom in... You'll get the picture: the function is very flat near x = 0. We will see what this means for the Taylor series approximation for the function f later on. Let us first concentrate on the derivatives of f for x ≠ 0. Since f is a composition of two differentiable functions, it is differentiable (for x ≠ 0). We get: f'(x) = (2/x^3) * f(x) if x ≠ 0 For the second derivative of f we find: f''(x) = (4*x^-6 - 6*x^-4) * f(x) if x ≠ 0 Let's do it one more time: f'''(x) = [ 8*x^-9 -36*x^-7 + 24*x^-5 ] * f(x) if x ≠ 0 (I hope I haven't made a mistake here) By induction we prove that: f(n)(x) = P(1/x) * f(x) if x ≠ 0 (*) Here f(n) denotes the n-th derivative of f and P is a polynomial (the degree of P is irrelevant for now). Note that the formula above states P(1/x), not P(x)! (for the 3rd derivative, the polynomial P(x) equals 8*x^9 - 36*x^7 + 24*x^5, but remember we have to plug in 1/x, not x) We MUST use the formal definition of the derivative to determine whether the first derivative of f exists for x = 0: f'(0) = lim (x→0) (f(x)-f(0))/(x-0) = lim (x→0) f(x)/x = 0 (**) since f(x) goes to zero much more quickly than x does if x→0. Since lim (x→0) f'(x) = 0, we now know that f' is continuous everywhere. Again, by induction, we can proof (using (*)), that all derivatives of f are continuous, differentiable and that f(n)(0) = 0 for all n. (again, we HAVE to use the formal definition of the derivative here as we did in (**)) So we now know that f is infinitely many times differentiable for all x and that f(n)(0) = 0 for all n. What does this mean for the Taylor series approximation of f? Well, actually it means that every finite Taylor approximation for f must be the polynomial Tn(x) = 0! Thus, here we have a (relatively simple) function for which it's Taylor series is NOT equal to the function itself: the function f is SO extremely flat near x=0, that only the zero Taylor polynomial/series can approximate it well. Not every function equals it's Taylor series: the fact that the function is infinitely many times differentiable is not a sufficient condition, neither is the fact that the Taylor series is convergent for all x... It needs to converge to f(x).

@JoseRodriguez-mz5zi 2 года назад

Math is nice and all but I'm waiting for you to throw the pokeball so we can battle lol jk

@michaelroyer818 3 года назад

As a former math tutor, I find that with the "harder" derivatives, I break this out and it allows me to solve the problem with ease. It also helps me to better understand limits.

@陈明年 3 года назад

Lets make all the cal1 questions definition of derivative then, I bet that student will rip the paper.

@blackpenredpen 3 года назад

Better yet, all epsilon delta proof 😆

@陈明年 3 года назад

@@blackpenredpen OK I will rip the paper then

@RobsMiscellania 3 года назад

To grow in mathematical maturity, it is necessary to look well beyond the equations. They fall naturally into place from the structure we are exploring: the mathematical scaffolding of the equations themselves. When I was 16, I was in such a hurry. I could certainly have been described as THIS student, and I just wanted the formula. Today, I struggle to make certain formulas stick in my memory, aside from the ones I learned much earlier in my youth. It leads to occasional moments of embarrassment in front of scholars smarter than I (by that I mean all of them); I mainly shrug it off, figuring I can derive them again from the landscape I've already explored, or look it up in a book. It's no problem for me. The equations come naturally from a deeper understanding of the structure I've studied. And more often than I realize, when I wander back through well-studied material, I find new perspectives to look at old things. What is old truly is new.

@ChristAliveForevermore 2 года назад

You perfectly encapsulated the art of mathematical study: we study the structure of truth, not the formulas which state a truth. The argumentative structure is what gives life to the mathematics. This fact is too deep for 99.9% of math students to realize until they get about half-way through a physics or maths degree, unfortunately. If we could teach children in Algebra 1 this fundamental and beautiful fact of mathematics, we may very well have millions more mathematicians and physicists than we do presently.

@colleen9493 3 года назад

You should re-name the video to: That one student

@omjoglekar3677 3 года назад

the question came to my mind but i dismissed it thinking that such a thing shouldnt exist because shortcuts will have to work always. but no !! Amazing video and concept. Oh..... and yeah my membership month is gonna run out soon . . . . gotta renew it asap !

@samuelatienzo4627 3 года назад

3:42 😂😂😂 he was just like “you get my point, there’s a shitload of rules, no need to list more”

@mathmathician8250 3 года назад

Hmm looks like I should include definition of derivative as one of the tricks to solve tricky derivative questions

@BroArmyCommander 3 года назад

Yes, you should! As did I

@yenyelinito 3 года назад

Very nice! I point out, I think there might be an editing error at 3:43

@rodrigolopez3874 3 года назад

Hi bprp, I've seen a recent video of Mathologer talking about calculus of differences, the ideas of calculus applied to sequences, where the integrals are sums and the derivatives are differences... That's so cool, I derived its "product rule" to difference and then the equivalent to integration by parts with the idea of get a closed formula for some sums like for example the aritmetic-geometric series and things like that... would be awesome if you can develop this ideas in a video! Much love from Spain

@blackpenredpen 3 года назад

Thanks. Have you seen the summation by parts? Prof Penn has a video on that ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-mNIsJ0MgdmU.html

@rodrigolopez3874 3 года назад

@@blackpenredpen Wow I didnt see it... that's exactly what I was thinking, thank you very much! This is so cool and even maybe can help me with some series problems

@applealvin9167 3 года назад

You know, that’s impressive…

@bobsanchez6646 3 года назад

Your example can still be solved just as easily with the product rule. Let g(x) = 2^x - 1, and let h(x) = (2^x - 2) ... (2^x - 9). Then f(x) = g(x) h(x). So f'(0) = g'(0) h(0) + g(0) h'(0). Since g(0) = 0, the quantity is just g'(0) h(0).

@GaryTugan 3 года назад

Very cool

@blackpenredpen 3 года назад

Thanks : )

@rikhalder5708 3 года назад

Please explain epsilon delta definition of limit ☺☺☺☺🙂🙂🙂🙂😊😊😊😊😊😊😊😊😊

@thaitrieu791 3 года назад

he did on another video

@ryuk2957 3 года назад

What if we take log(.) On both sides and then differentiate it. F'/F = ln2 (2^x) /2^x-1 + .......... Now F' = ln2 (2^x) {(2^x-2)(2^x-3) ..... (2^x-10) + ........... Now At x=0 Expect first term other terms will be 0. And first term will be -ln2 (9!)

@danlogic465 2 года назад

Difference quotient fan vs Power rule enjoyer. Anyways, the video was well explained. I still hate myself everytime I do the difference quotient.

@tambuwalmathsclass 3 года назад

Am I the only one who enjoys this tutorial?

@aaron62688 2 года назад

how I personally teach it is to guide them through deriving (inventing) the definition of derivative themselves. That really drives home the meaning. Then after discussing it philosophically for a while I go through and walk them through deriving some of the rules. How many of them depends on time available and if they seem into it. But I always go through the derivation of the power rule using Pascal’s Triangle. Ill occasionally throw them a problem where the instructions are to use the definition but I never beat them over the head with it. Its disingenuous to do so. And the classical teaching method of doing a mega crap ton of problems with the definition to only then later show them the shortcuts really pisses students off. And there is no reason for it to be done that way! Im honest from the start and tell them that we are deriving a definition that can be used and may be useful sometimes but that there are shortcuts available down the road. They trust me to not bullshit them so when I say “lets take a look at the definition and discuss it then do some just so you can see it work” they respond positively. And when I am inevitably asked “when will I ever need to use the definition to do a derivative?” I answer “Probably never. But we aren’t looking at it for the sake of using it. We are looking at it because we want to understand what is happening behind the curtain. And ill give you a few reasonable ones to do just for practice in the case you do end up needing it and so I can see you weren’t asleep this whole time”

@heil95_52 2 года назад

Well actually here in France the teachers do it the easy way :/ And we got tons of different formulas to learn for different derivatives.

@joaomatheus6222 3 года назад

3:44 He Just gives up

@rathnakumaran673 11 месяцев назад

2:25 "Impressive I know this too, you know?" Cold, cold as ice

@potato_cm 3 года назад

As a physics student I have needed to use a variation of the definition of the derivative to derive certain equations in classical mechanics.

@Thesupremeone34 2 года назад

Ah yes Open Matlab and load in the forward finite difference algorithm you had to write in like physics 105

@BorbasTV 3 года назад

h/h doesn't "Cancel Out," it reduces to one. For instance in xy = yz, you can't "Cancel Out" the y because you can only "Cancel it" if y doesn't equal 0. So, saying "Cancel Out" is actually informal and saying reduces to one lets the reader/student know that the y doesn't just disappear yet that it goes to a value of 1. My teacher was always super rigorous when it came to math.

@SmallSpoonBrigade 2 года назад

That's nonstandard. h/h has the hs canceling, as the term is generally accepted to mean. You can cancel them because there's a removable discontinuity that we've already decided to ignore via the limit pushing the sides together. You can definitely cancel the y in that expression, you just have to be mindful about the possibility of y = 0. In that case, it doesn't much matter what x or z are equal to, as any value of either would satisfy the equality. So, you'd likely set a domain restriction on y and move on.

@Garlic5auce 2 года назад

I know this too, you know??!!😂😂😂

@AnakinSkywalker-zq6lm 3 года назад

This is very useful because I have absolutely no idea what derivatives are! I’m only learning limits now

@monsieurdidkekne3224 2 года назад

You don't need the derivative definition to derive 2^x you transform it into the definition using 2^x=exp(X(ln(2)) and that's easy to differentiate.

@jamesflanagan6977 2 года назад

Is there a problem where it would be easier to use the definition of multiplication (a*b = Σ(n=1,b) a) rather than multiplication rules? That's what this question sounds like. Knowing definitions might not be as practically important as learning how to use them but it's still an important part of understanding

@chimetimepaprika 2 года назад

Doing derivatives on the streets could save your life. You never know.

@AndDiracisHisProphet 3 года назад

2:08 easy solution: just give that student a good slap on the head! that should fix it

@sploofmcsterra4786 2 года назад

In physics we also "discover" derivatives. E.g. If you are considering the pressure on a horizontal slice of fluid in a vertical tank, you end up with something like P(z+h) - P(z) = gravity*density*h. Then you know that if you divide both sides by h, the left hand side is just dP/Dz, so the pressure is the integral of gravity * density!

@sploofmcsterra4786 2 года назад

(h is just the height between the slice we are considering and a point a tiny distance above it.)

@megaldon1086 Год назад

Hey, I've just discovered your channel and it's very cool. I've just finished what would be grade 11 in Spain (1r de Bachillerato) and I must tell you that I have learned to do the derivatives of functions using formulas for each case instead of using the general formula. For instance I have written down that the derivative of loga(x) = 1/(x*ln(a))

@teslachess7319 3 года назад

Is it possible to make a definition of an integral by the definition of a derivative?

@abhishekbanerjee4458 3 года назад

Are you challenging me to not use the product rule on the last question?

@ctendstoinfinity 3 года назад

Can you integrate 2xdx with lim-> 10 to 13

@benzel5659 2 года назад

3:12 yeah I too get asked to differentiate something in the street all the time.

@hellNo116 2 года назад

I am trying to become a professor at a university level as well. You videos are so motivating and relaxing at the same time

@GPLB 3 года назад

“The proof of this is left to the reader.” 😂😂

@MM3Soapgoblin 2 года назад

As a physicist, you almost never solve for a derivative using the definition (I at least have definitely never) BUT the definition is very important because there are several cases, classical mechanics and statistical mechanics especially come to mind, where you are solving a physical problem and end up with the definition of a derivative. If you never learned the definition you would not be able to solve it beyond that point. This is true in general rather often, where you rarely use the definition or derivation of a method to directly solve problems with that method but knowing the derivation lets you recognize it and transform your difficult problem into one that you already know the tricks to solve.

@Seba-xs8bb 3 года назад

its impressive seeing how someone can just do calc math

@neilgerace355 3 года назад

3:46 Congrats for starting Take n+1 but not erasing Take n.

@blackpenredpen 3 года назад

Lol

@janhetjoch 2 года назад

I'm in highschool and my mathbooks and teacher just thought me the specific rules (like the power rule and multiplication and division rules and such) but never the general formula of making a derivative. I prefer knowing the formula over specific working outs of the formula even if I'll usually will still use the rules. So thank you

@TheMuffinMan 2 года назад

My calc teacher really spent days going over this only to hit us with a hammer and tell us we bring the exponent down. Now here I am going back to square 1 but with polar coordinates in calc 2

@SimsHacks 2 года назад

In real analysis, you just prove all the derivative rules using limit definition, so you don't use it for concrete examples then.

@mr_angry_kiddo2560 3 года назад

Sir you are fan but Iam lover 😍😍

@theGoogol 2 года назад

I'm studying Python and am being confronted with calculus. Thought I could learn something here. But this is way too fast and too advanced for me but I admire skill so this vid was enjoyable.

@horadetodososlegos 3 года назад

Why the thumbnail says you're a fan if you're not spinning? Right, I'm leaving

@laucheukming5580 3 года назад

I reckon that your 2nd example should apply linear approximation as you let f'(x) = f'(0). It seems like you take derivative of this function at x = 0, and you guys should recognize this approach should be more easier to handle since it will appear a bunch of zero. Remarks: It is just useful when x is really really close to zero but not x=0. According to def. of derivative, it just apply when one of the two points is close enough the other point.