easy derivative but it took me 32 minutes

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We will find the second derivative of sin(x^2) but without the chain rule, power rule, and product rule. Instead, we will just need ONE limit! This is a hidden gem for limit enthusiasts! The key is to use the limit definition of the second symmetric derivative to find the second derivative of sin(x^2). Note, that if the function is second-differentiable in the usual sense, then the second symmetric derivative is equal to the regular second derivative. Since sin(x^2) is always differentiable, so let's find d^2/dx^2(sin(x^2)) with this rarely-taught approach. We will also need two famous limits, the limit as x goes to 0 of sin(x)/x and the limit as x goes to 0 of (1-cos(x))/x.
0:00 finding the second derivative of sin(x^2) but no differentiation shortcuts
0:54 Definition of derivative, symmetric derivative, and second symmetric derivative in terms of limits
7:46 The real action!!
9:08 The formula for sin(A)+sin(B)
12:01 Back to the limit
29:50 Check the answer!
Check out my 100-limit video for more calculus limit practice. • 100 calculus limits (f...
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Thank you all!

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2 июн 2024

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Комментарии : 258

@blackpenredpen 6 месяцев назад

If this one limit isn't crazy enough, then try 100 limits: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-TglD4Y6lmQk.htmlsi=lR_jfa-gI8FVO7VS

@muddle. 6 месяцев назад

already done sir.

@Mandq. 6 месяцев назад

easy 🙅🏼‍♀

@francorota8638 5 месяцев назад

This is the first time I was introduced to symmetric derivatives, as my college professors never taught me about it.

@mathmachine4266 6 месяцев назад

Unfortunately, this problem is unsolvable. Not because there is no solution, but because my board isn't big enough.

@user-gs6lp9ko1c 6 месяцев назад

Why, it's a one line derivation! If you start sufficiently far to the left. 🙂

@unanimatereactor5014 6 месяцев назад

@@user-gs6lp9ko1cas long as the size of your letter approaches 0

@senseof_outrage9390 6 месяцев назад

I have discovered a truly marvelous demonstration of this proposition however this whiteboard is too small to contain.

@thexavier666 4 месяца назад

Assume you have an infinite whiteboard

@unanimatereactor5014 4 месяца назад

@@thexavier666 no it tends to infinity

@YoungPhysicistsClub1729 6 месяцев назад

Bro really used the limit definition, legend status

@blackpenredpen 6 месяцев назад

😆

@thomasblackwell9507 6 месяцев назад

That is a fact!

@bryantwiltrout5492 6 месяцев назад

Pure madness too 😂😂😂 using the Limit Definition to do derivatives of Trig functions is crazy 😂😂

@andreasxfjd4141 5 месяцев назад

Even worse to integrate with series

@lukeparsons4965 4 месяца назад

I’m just gonna do a reihman sum with infinite rectangles

@mcalkis5771 6 месяцев назад

Now you have to do an epsilon delta proof of the limit for the ultimate presentation of mathematical rigor.

@blackpenredpen 6 месяцев назад

😂

@yttyw8531 6 месяцев назад

but before that you need to proof that 1+1=2

@user-cm5qn1fq3e 6 месяцев назад

This would take 12 hours but would be a good video

@Kanin105 6 месяцев назад

Xddddddddddd

@adb012 5 месяцев назад

@@blackpenredpen ... Why do you laugh? It was not a joke.

@apolloo9068 6 месяцев назад

It's reassuring that I'm not the only one prone to making calculation errors. Great video!

@Romeo-qk8tk 6 месяцев назад

As an aspiring student in AP Calculus, this video was incredible to see! Awesome content! ❤

@blackpenredpen 6 месяцев назад

Thank you!

@Sir_Isaac_Newton_ 6 месяцев назад

You're gonna fail AP Calculus blud

@harrymetu2746 6 месяцев назад

@@Sir_Isaac_Newton_😂😂😂😂

@harrymetu2746 6 месяцев назад

@@Sir_Isaac_Newton_when Newton says it 💀

@obi-wankenobi8413 5 месяцев назад

@@Sir_Isaac_Newton_I just got an A on my semester final it’s really not that hard if you pay attention (assuming good teacher)

@paytonholmes6019 6 месяцев назад

I don’t think I was ever taught what a symmetrical derivative is in my calculus classes. Thank you.

@fabianwho9797 5 месяцев назад

Im no expert, but in my judgement it is rarely useful for anything, so most people never hear of it

@Frankie18O4 5 месяцев назад

@@fabianwho9797 you can use it in numerical applications as a second order approximation of the derivative: [f(x+h)-f(x-h)]/(2h) = f'(x) + O(h²) in contrast to [f(x+h)-f(x)]/h = f'(h) + O(h) (for h→0, assuming f∈C²)

@EmpyreanLightASMR 5 месяцев назад

I'm still a bit lost as to why he had to go that route. Maybe he said why and I missed it. The symmetrical derivative can be used for a symmetrical function about a non-differentiable point, I guess. But why does he use it for sin(x^2)?

@bred223 4 месяца назад

@@EmpyreanLightASMRusing the normal def of deriv i got it in about 7 minutes so i assume it was just for funsies

@maxmustermann3938 2 месяца назад

@@fabianwho9797 it is very commonly used for numeric derivatives (Central differencing), especially on i.e. images when computing gradients or laplacians or calculating the slope or the normal of a heightmap, also heavily used to solve grid-based fluid simulations

@happyhippo4664 6 месяцев назад

I am a 64 year old chemical engineer, still working. Math has always been my strongest subject. I enjoy these videos very much. I feel if you do not understand math that well, you will have lot more difficulty in engineering.

@blackpenredpen 6 месяцев назад

Thank you!!

@mesindetrabajinv666 6 месяцев назад

Can you give me some advice for college and engineering? I’m planning to study Aerospace engineering

@Scorik375 6 месяцев назад

and even in many aspects of life

@happyhippo4664 6 месяцев назад

@@mesindetrabajinv666 Look at the occupational handbook for job outlook. Chem Engineering was hard but I've heard aerospace is even harder. My concern is, unless you are exceptional, it may be harder t find a job in that field. I started in Chemistry but switched to Chemical Engineering when I found out that BS ChEs were getting almost same pay as PhD Chemists. Probably more important, do what you enjoy.

@exodiara6392 6 месяцев назад

Im not so sure. Im not applying so much advanced mathematics again in carrier. Even friends that were more advanced had forgotten linear algebra.

@laurensdehaan2202 5 месяцев назад

Man, your enthusiasm is SO contagious! I stood up here in front of my computer and watched the whole thing straight through, with a couple of pauses to reassure myself why some things worked out the way they did! Thank you!

@EmpyreanLightASMR 5 месяцев назад

Same! My attention span is such that I can watch a full movie over the course of several days. I saw this video and thought I'd kill a few minutes before going to make dinner and ended up watching the whole thing. I was rapt!

@a-manthegeneral 6 месяцев назад

13:05 I'm a CS major (junior btw) watching this lol These videos make me feel good lol

@dinohuntr851 2 месяца назад

You have become my favorite RU-vidr. Your teaching style is fun, you aren't afraid to show us your mistakes, and you are just enjoyable to watch. I can tell you genuinely want to teach, not just show off your skills. Keep up the good work!!

@jaysonbunnell8097 6 месяцев назад

This was super awesome! I took calc 1 in highschool, and I've taken calc 2, Differential Equations, and Linear Algebra in college. I don't have many math credits left to take, so I find these videos awesome for keeping me on my math skills. Thank you!!

@slytherinbrian 6 месяцев назад

This is better than anything on netflix!

@jimschneider799 2 месяца назад

I'm neither a student nor a teacher. I'm just an old fart engineer who loves math enough to realize I really let myself get rusty on the basics. So, although most of the math I do at work is related to number theory, I do appreciate the refresher, particularly since you tend to tackle problems in ways different than what I would use.

@JoaoVictorCavalcanteMiranda 2 месяца назад

Thanks for the great video! I didn't know about symmetrical derivatives until now! Your excitement is worth of the challenge!

@Prism019 6 месяцев назад

31:38 You can hear the relief in that "Yes!" Congrats on getting a good take!

@mcalkis5771 6 месяцев назад

Always a good day when you upload Steve. I always enjoy your videos where you do proofs like this.

@blackpenredpen 6 месяцев назад

Thank you so much!

@argonwheatbelly637 6 месяцев назад

This is math candy. Awesome! ❤

@user-xm6ev1tv1i 6 месяцев назад

Sinx^2을 두번 미분~ 첫번째미분 : 2xcosx^2 두번째미분 : (곱의미분적용) 2cosx^2 - 4x^2sinx^2

@ivantolkachev4808 6 месяцев назад

For cos(2xh) - 1 you can also do the cos^2(xh) - sin^2(xh) -1 = -2sin^2(xh) to avoid the trig in the denominator

@garyhuntress6871 6 месяцев назад

That was excellent!! I was on the edge of my seat!

@gallium-gonzollium 6 месяцев назад

Me when I try a calculation and I do it the more complicated way:

@Dantido 6 месяцев назад

Hey there. Before asking this, just wanted to say I love your videos. Thanks to you I've found out about my interest in math as a hobby, and I can't commend how interesting and satisfying stuff like calculus can be when you understand it enough. With that out of the way, I also wanted to ask you a question, which definition of derivative do you prefer? f(x-h) - f(x) lim ---------------- x->0 h or f(x) - f(h) lim ------------- x->h x - h

@m1n3c4rt 6 месяцев назад

wow, your videos are so consistent that i didn't even notice this was from an hour ago also 25:30 funny integral sign :)

@TheZerovirus1000 6 месяцев назад

ikr! I didn't notice until you pointed it out. I love this format

@albertogarcia4177 6 месяцев назад

Seems all ok but this finding of the second derivative using the symmetric derivative as shown is good only for x≠0, see 26:57 where you multiply and divide for 2x. Would be need complete the proof for x=0, i guess is not hard, pluging x=0 in the initial steps, see the formula is also ok

@darktrinity9125 6 месяцев назад

Now prove that limit is true by the definition of a limit (epsilon delta)

@tortillajoe9942 6 месяцев назад

Now do ε-δ 👀

@anandmoodley3787 6 месяцев назад

That was brilliant!. As a student I enjoyed every bit of that

@trelosyiaellinika 6 месяцев назад

Absolutely beautiful!

@BradMurray 2 месяца назад

This was really beautiful; thank you!

@Veefencer 4 месяца назад

The only thing I like more than trying to follow all of the steps you take during the video is your sheer joy when the thing is finally done! I can't help smiling as well. Thank you for reminding me that i really like math!)

@kooshkooshyunger1438 6 месяцев назад

You forgot to close the parentheses of the first d/dx at 30:27

@gajamsai2957 3 месяца назад

31:36 the excitement and happiness 👏😘 such happiness in his face very nice to see good answer sir 👍👏👏

@Owen_loves_Butters 6 месяцев назад

18:53 Actually, you can use L'Hôpital's as long as you don't derive d/dx[sinx]=cosx from it, and there are other ways to prove the derivative of sine is cosine :)

@velimir_ikalovic 5 месяцев назад

What amazes me is that I'm 47 yo, finished my highschool long time ago, dropped from university on second year, I don't use calculus in my life at all, and I still manage to understand most of this.

@punitpasricha3876 6 месяцев назад

We extremely appreciate your effort

@nathanperkin1163 6 месяцев назад

I'm 15 and in year 11 (grade 10), and i haven't officially been taught calculus yet, but i find these kinds of videos super interesting!

@lirosphere956 6 месяцев назад

You're in for a treat if you go deep in this channel

@hattapalkan8395 6 месяцев назад

Proud of you brother. Keep it going

@idjles 6 месяцев назад

Calculus will keep you fascinated for the rest of your life - even when you are 80. Keep enjoying it.

@TheEGod. 6 месяцев назад

im a little younger then you but i underatand very well. I remember I didnt understand them a year ago but I just couldnt stop watching these videos.

@diomidis.nikolaou 6 месяцев назад

same here bro

@stolenmonkey7477 2 месяца назад

1:53 this was genuinely so funny I love that lol

@johnchestnut5340 Месяц назад

I do appreciate it! Thank you. I am surprised at how much I remember. I am also keenly aware that I have forgotten so very much. But thank you for the videos!

@spudhead169 6 месяцев назад

You've explored tetration a few times, but can the concept tetration be extended to include non-integers? If we define Tn(x) to be a tetration function such that T3(x) = x^x^x and T5(x) = x^x^x^x^x, then what happens if we plug in say 2.5 for n or even i? Is that even possible?

@ciple8330 6 месяцев назад

I researched this a bit and it seems very interesting. For anyone interested, go to Wikipedia>Tetration>Extensions (go to the 'heights' section for n) and search up "Tetration Forum" if you want to see more discussion on tetration.

@HasanZekiAlp 6 месяцев назад

Hey, Guy, you’re great! While I was following you, frankly, I lost a few times, but, you delivered the ship to the quay, in sane!…

@alexprospal8548 5 месяцев назад

I apreciate it. Thank you. We all apreciate you.

@askandpushpaltiwary8537 6 месяцев назад

appreciate the hard work!

@dudl2945 6 месяцев назад

I'm always looking for entertaining videos to watch while eating my food. Although I love maths, I would have never guessed I'll end up with this kind of stuff as best eating videos

@oryx3 6 месяцев назад

I think we're going to need a bigger board 🦈

@kevinstreeter6943 6 месяцев назад

Reminds me of the time when I was asked if there is 2 ways of solving a problem, which one should be used? I replied to use the easier method.

@joshuahillerup4290 6 месяцев назад

I can't wait for the follow-up video

@aymathconcoursprep 3 месяца назад

this derivitive AkA fs(x) used also in Mechanics Element finite or Numerical methode to describe descritisation of grid points in the plan

@michaelroy1631 6 месяцев назад

very satisfying!

@General12th 6 месяцев назад

So good!

@DarkTouch 3 месяца назад

im never going to use the product rule and chain rule again because i loved doing it the long way. !!!! Nice proof.

@SakretteAmamiya 5 месяцев назад

for the cos(2xh)-1, you can make it into -2sin(xh)^2 😀

@felipesb2 6 месяцев назад

31:22 Hahaha I thought that I was the only one who get that excited when I finish a "uncrackable problem" like this one good job

@lol1991 6 месяцев назад

I literally solved the first derivative of this last saturday (by the definition)

@scottleung9587 6 месяцев назад

Very cool!

@alexdefoc6919 6 месяцев назад

i can aprove this worked. 30:08 before this. In my head ❤. I am really happy i can do it.

@Annnabannanna 6 месяцев назад

Omg, grea ttiming, I have an exam in 2 days on calculus.

@WingedShell82 5 месяцев назад

That was a very fun 32 minutes of my life. Honestly, this makes me want to practice doing this kind of stuff on my own just because it looks fun lol.

@EmpyreanLightASMR 5 месяцев назад

It's fun when you are seeing the steps you have to take. Not so much when you're standing at a crossroads. Though I learned earlier today that sometimes you just have to make a decision (a big decision, as Bob Ross would say) and stick with it until you hit a dead end.

@lornacy 3 месяца назад

I will have to watch this more than one time!

@joshuahillerup4290 6 месяцев назад

It's been a long time since I took calc 1, but I'm now getting flashbacks to a bunch of weird limit definitions

@xyz.ijk. 6 месяцев назад

Chain rule? What is the Chain rule? What happened to Chen lu? Did you mean Chen lu? I miss the old days. You are still a great teacher and always had a wonderful sense of humor.

@lukaskamin755 2 месяца назад

5:30 and what if the graph is not symmetrical relative to the "angle" point, say if there's a sum or difference of a linear function and an abs function with various angle coefficients. What meaning does symmetrical derivative has, will it be relevant? It doesn't seem obvious

@shubhamsaini4871 6 месяцев назад

How to check if we can integrate a function or not?. Like e^cosx is non integratable so how do we know that it is non integratable

@earl8295 6 месяцев назад

Blackpenredpen be like: Im going to prove this using Epsilon Delta defintion

@blackpenredpen 6 месяцев назад

I will pass this time 😆

@natural-log-here 6 месяцев назад

Try e^x + lnx = 0

@Yougottacryforthis 6 месяцев назад

Nasty. How do you solve that other than numerical estimation?

@natural-log-here 6 месяцев назад

@@Yougottacryforthis I want to see if bprp can get exact form like x^x = 2 Normally it’s just numerical estimation but he used Lambert W. Can he use the lambert w or other ways to do the request in exact form?

@AshXcod 6 месяцев назад

14:56 (x+h)^2 = x^2 + y^2 + 2xh, (x-h)^2 = x^2 + y^2 - 2xh, select alpha=x^2+y^2 and beta=2xh and you didn't need the sum to product formula :)

@sphakamisozondi 6 месяцев назад

Bro did derivative calculus using 1st principles. I'm beyond impressed

@indescribablecardinal6571 5 месяцев назад

26:55 When both limits needed that 2x•2x/(2x•2x), and that finally connected with the answer given by chen lu, then I smirked a little c:

@gietie1694 6 месяцев назад

its beautiful

@potato4521 4 месяца назад

I was able to do it by applying the derivative twice so d/dx (sin(x^2)) = cos(x^2)*2x. Then d/dx (cos(x^2)*2x). using the product rule you get -sin(x^2)*2x*2x + cos(x^2)*2. This simplifies to -4x^2*sin(x^2) + 2*cos(x^2)

@theweirdwolf1877 4 дня назад

No shit sherlock but that’s not the point of the video

@charlievane 6 месяцев назад

31:36 happiest man alive 😂

@genocider5868 6 месяцев назад

wait, does it mean you can just use symmetric derivative as a normal derivative? Or is it only second symmetric derivative = second derivative? Or is it just coincidence?

@blackpenredpen 6 месяцев назад

If the function is differentiable in the usual sense, then yes. Symmetric derivative = regular derivative

@Memzys 6 месяцев назад

you can use symmetric derivative definition in place of the regular derivative, they should be the same, except sometimes the symmetric derivative will you give you an answer while the regular derivative does not (when there is a sharp turn in the graph, like f(2) = |x - 2|) quoting wikipedia, "If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true" this applies for any nth derivative, not just the 2nd

@jacquesduranceau8762 5 месяцев назад

My students used to keep a catalog of the outrageous things I said over the year. I think "just give up" and "don't be happy to quickly" satisfy the criteria!. I have one question: Doesn't the "'symmetric derivative" (which looks akin to a central finite difference) lead to f' = 0 for absolute value at the critical point?

@eggthepro2472 6 месяцев назад

17:07 After the 4th line, I know why he factored out the 2sin(x^2) but i don’t necessarily get how. The 2nd term in that question doesn’t have the factor sin(x^2)?? am i just reading it wrong Also, I don’t understand why he uses the symmetric derivative for this function to begin with. I don’t quite know if it’s easier, but the function itself has a derivative with the domain of all real numbers, so wouldn’t it be easier to just take the derivative of that again?? i think he said that the symmetric derivative doesn’t imply the existence of the first derivative but if the first derivative does exist then what’s the point?

@jbucata 6 месяцев назад

It's not the 2nd term in the numerator, but the 3rd term (which is where the "-1" then came from). As he explained at the beginning, if you know the normal derivative exists, then the symmetric derivative also exists, and the two are equal. Since you can choose either representation, evidently the symmetric derivative was easier to work with.

@eggthepro2472 6 месяцев назад

@@jbucatahaha i totally just didn’t see that at all, thanks for pointing that out

@lore7_03 5 месяцев назад

"i'm not gonna prove that limit, I'm just gonna use it" proceeds to prove that limit

@black-shadow5715 6 месяцев назад

Can we use the following definition to calculus the second derivative of sin(x²) ? f''(x) = lim h->0 [ (f(x+2h)-2f(x+h)+f(x)) / (h²) ] I don't know if using this formula would be more easier than the other one in the video, I don't have the energy to try it 😂 And also, I don't know if the formula is right or not but I do a little demonstration to proove it and I think the formula is right... Please tell me 👀 PS : sorry for my english, I'm actually french

@thenew3dworldfan 6 месяцев назад

Definitely, in fact this might be there regular second derivative not the symmetric one, which might exist even if the function isn’t twice differentiable?

@wentianxiang1155 5 месяцев назад

Great teacher

@TheZerovirus1000 6 месяцев назад

31:37 certified mic drop / pen drop moment

@lautamn9096 6 месяцев назад

You're amazing

@rockapedra1130 2 месяца назад

This was endless excruciating pain followed by revelation. Sometimes you have to have faith and keep slogging! Wow! You did this as an 8 year old? Proof of alien visitation! I'm calling the History Channel!!!!!!

@Diro_Nikhil 6 месяцев назад

Man never used the 2nd derivative by 1st principal…. This is so good

@EmpyreanLightASMR 5 месяцев назад

Everyone is inspired by this great video, but apparently I'm the only one who heard two cries for help. Twice! you asked for a bigger house to fit a bigger board. Someone get this man a larger house!

@blackpenredpen 5 месяцев назад

Thanks! I appreciate that. 😃

@user_cy1er 6 месяцев назад

Watching this as a 7th grader just feeling cool😎

@YoungPhysicistsClub1729 6 месяцев назад

if you understand this, I will be shocked

@user_cy1er 6 месяцев назад

@@YoungPhysicistsClub1729a bit i guess,just didnt learn about trig(are they sin cos tan cot and other 2)

@windowsxpmemesandstufflol 6 месяцев назад

Remember to use the chen lu

@shanathered5910 6 месяцев назад

@@user_cy1er sec and csc

@Furetto126 6 месяцев назад

Imo calculus is one of the things that can be pretty easily understood by everyone that likes maths, but it's actually hard to know how to solve problems that use it

@user-bf3ko7ts5e 6 месяцев назад

It's simple. Already the first derivative of this function “jumps” very much. For x> 2pi for example.

@bobbybannerjee5156 6 месяцев назад

So the regular and symmetric derivatives are totally different things right?

@iandress1 6 месяцев назад

🎉🎉🎉 thanks you communicate blazing good 🎉

@probablemente_sebas6612 5 месяцев назад

OH MY GOD we had the exact same expression of the symmetric derivative as homework and the teacher never told us it was an actual definition 😭.

@ThsHunt 2 месяца назад

When teacher asks a simple question for 10 points

@Amoeby 6 месяцев назад

When I saw that we need to find (sin(x^2))'' I was thinking how is this video this long and then BRP pulled out the limit definition of the derivative.

@Deniz-zd2nx 6 месяцев назад

Can you find the complex solution of 3^x+3=30 of course if there is

@xxxgetr3ktxxx740 6 месяцев назад

3^x + 3 = 30 3^x = 27 x = log_3(27) x = ln(27)/ln(3) x = 3ln(3)/ln(3) x = 3 Solving the normal way, we'd get only 3 but we can get the complex solutions when using Euler's formula to get 27 = 27e^i2nπ Plugging back in: x = ln(27e^i2nπ)/ln(3) x = (ln(27) + ln(e^i2nπ))/ln(3) x = (3ln(3) + i2nπ)/ln(3) x = 3 + (2nπ/ln3) i, n∈Z

@sadi_supercell2132 5 месяцев назад

1:53 i laughed so much i almost died hahahaha , " lets just give up " haahahahah thank you 😂😂😂😂

@Heramb_Pathak5879 6 месяцев назад

Wait I have a doubt: If we have taken the limit only one time, how is this the second derivative of the function?

@dihydrogen 6 месяцев назад

that limit finds the symmetric second derivative of the function.

@major__kong 6 месяцев назад

What the other poster said. He skipped a step in deriving the formula for the second derivative and just gave it to you. But all you do is apply the symmetric derivative once to get the first derivative then apply it again to the first derivative. If you want to see the derivation, search for centered finite differences.

@NintendoGamer789 6 месяцев назад

Basically you can guarantee both h go to 0 synchronously by MVT

@Mandq. 6 месяцев назад

if anyone used chain rule and producst rule the answer would be: 2( -x^3 * sin(x^2) + cos(x^2) )

@Prussiluskan42 5 месяцев назад

Awesome 👍

@guy_with_infinite_power 6 месяцев назад

Hello sir

@courageouscuber9278 6 месяцев назад

13:11 "If you're also a calculus teacher, you know what to put on the final exam" Don't give my professor any ideas 😂

@Scorik375 6 месяцев назад

do you know that derives as you write do not mean derives in mathematics , it means derives in physics , but to calculate a derive in mathematics can be only count with limit

@mohamed.chakib_ 6 месяцев назад

100 double intégral and 100 triple intégral ❤

@toopytoopy8547 5 месяцев назад

maybe using the fact that sin(x²)=Im(exp(ix²)) helps as d²sin(x²)/dx² = Im(d²(exp(ix²))/dx²) And with your method, exp(i(x+h)²)+exp(i(x-h)²)-2exp(ix²) = exp(i(x²+h²))[exp(2ixh)+exp(-2ixh)]-2exp(ix²)= 2exp(ix²)[cos(2xh) exp(ih²)-1]. Then, we have [cos(2xh) exp(ih²)-1]/h² = exp(ih²)[cos(2xh)-1]/h²+[exp(ih²)-1]/h². For the first part, we know that lim [cos(2xh)-1]/h² = -4x²*lim [1-cos(2xh)]/(2xh)² = -2x² and lim exp(ih²) = 1. For the second part, lim [exp(ih²)-1]/h² = i (as h²->0 and lim [exp(ih)-1]/h=iexp'(0)=i). Thus, lim [cos(2xh) exp(ih²)-1]/h² = -2x²+i . Therefore, d²(exp(ix²))/dx²=(-4x²+2i)exp(ix²) and finally, by taking the imagenary part, d²sin(x²)/dx² = -4x²sin(x²)+2cos(x²)