checking the answer of integral of sqrt(tan(x)) by differentiation

Most difficult derivative question on my channel yet,
checking the answer of integral of sqrt(tan(x)) by differentiation,
hardest differentiation problem,
math for fun,
hard calc 1 derivative problem,
blackpenredpen

[@tylerqueen8920]

It's just so beautiful...*tear falls slowly from eye*

[@blackpenredpen]

Tyler Queen you won the comment of this video!! I just pinned it!! What a wonderful comment!!

[@pongwang9893]

@@blackpenredpen yeah , so brilliant . great

[@AP12uZvfvag]

Last time someone used chain rule so many times, slavery was still legal.

[@afsarshaikh9654]

AP12uZvfvag You made me laugh. thank you.

[@megauser8512]

lol

[@BizVlogs]

ummmm… THE FUCK?!

[@donkeybros8734]

I’m dead 😆 😂

[@erikmensinga]

No its the chen lu

[@stuartyellow1679]

Why you dont call the video "differentiat (-1/sqrt(2)*tanh^-1((sqrt(tan(x))+sqrt(cot(x))/sqr OK i got it

[@blackpenredpen]

exactly

[@diegobatista2239]

latex gang

[@joelottosson1297]

You like when they cancel out don't you

[@blackpenredpen]

sure i do!

[@ZoneEEEEEEEEEEEE]

It's just so wonderful, it's almost like magic :')

[@bastianherrera2171]

I've watched most of your videos and they are amazing, mainly because of the clear way you solve the problems. I've learnt so much about calculus with you, thank you very much! Keep making these amazing videos!

[@blackpenredpen]

Thank you for the supporting message! Will do!

[@morganthem]

blackpenredpen I second this, they are amazing and clear

[@AhnafAbdullah]

I did this derivative after he put out the sqrt(tan x) but before this video, it was so beautiful :-:

[@JeronimusJack]

oh yea that's excatly how i feel, too. my old math tutors always wanted me to break more about it instead of telling me how it's done, but sometimes i was just stuck, here i get answers. i did the derivation afterwards myself :D and it worked. it's better to learn this way since i can explore and my stress level stays down hehe. it's not torture but tutoring here. ty

[@tcocaine]

I love how in the thumbnail you just wrote craziness instead of the whole shablang. I absolutely adore it.

[@blackpenredpen]

HAHAHA, thanks!!!!

[@agabe_8989]

That beast was too long that he had to type "integral of sqrt(tan(x))".

[@whiz8569]

"Don't be intimidated cause I did 1+1=2 again." I see what you did there.

[@blackpenredpen]

whiz 85 hahahahahahaha!!!!!

[@mrbenwong86]

After 15 minutes, I forgot I am doing differenitation, I almost want to put +c at the end.

[@throxs1535]

I feel like you're really getting the hang of this now :) going slow where need be... Going faster where allowed... This all comes together in a very fluent yet easily understandable explanation :) keep it up!

[@blackpenredpen]

Throxs MC thanks!!!! I think it's bc I have been just doing higher level math lately so it was easier to get the pace right.

[@DancingRain]

That was epic! The most amazing mathematical simplification I've ever seen.

[@dizilloHT]

wow finished my calculus class this year, and i dont believe i will see it again, but wow im so amazed by it. Keep up with the videos you help a lot of people with them!

[@turkishwagnerian]

"I chose mathematics to honor the human mind" Carl Gustav Jacob JACOBİ

[@morganthem]

Wow

[@soyoltoi]

tanx for the video

[@user-mt9ux2di6u]

I really love how it all just collapses down to sqrt tan x

[@Sid-ix5qr]

d/dx(craziness) Let's do L-sub. l=cr d/dx(laziness).

[@weerman44]

Yet again, an amazing video with such a clear explanation

[@Magic73805]

Really, You're great sir. You are the master of "Calculus".

[@azmath2059]

That's a great deal of work! Thanks for posting. Brilliant

[@kornelviktor6985]

Its so nice to see that he gets very excited when he sees that the answer will be sqrt(tanx).

[@furo822]

I think this is better. For the 1/(tanx + cotx) Multiply both sides with tanx ->tanx/(tan^2(x) +1) and the bottom part becomes sec^2(x) Finally everything just cancelled out and sqrt(tanx) remains That was funny btw

[@blackpenredpen]

YungHong Kung ahhhhhhh that's better!!!

[@flowingafterglow629]

So I watched the integral of root(tan x) the other day, and therefore had to watch this. I like the way you were smiling through the whole thing. I was just laughing the whole time, it is so beautiful to see. Every time something would fall out, I would just laugh some more. Amazing.

[@MRBYRNO2505]

holy mother of god, if i saw this a few years ago i would not have chosen to study maths at Alevel. seeing it now just brings a tear to my eye its so beautiful

[@tomatrix7525]

That is a beautiful piece of art my friend

[@samirgeiger1042]

This is just so amazing. Great work! Such beutiful derivatives

[@Galileo2pi]

Great, keep going please

[@bartj777]

Bravo, sir. Very impressive.

[@ttg001]

I'm holding back tears....... This video is so powerful

[@AlmightyMatthew]

just perfect. satisfaction at the end was through the roof

[@drewski91]

i started crying at the sheer beauty of this math

[@snejpu2508]

0:05 "We'll be differentiating this chinese expression". : )

[@buckeye49]

He said giant

[@BlokenArrow]

That’s wasist

[@dakotaroberson9921]

Wow great proof!! it was so great to see it all come together at the end lol

[@blackpenredpen]

I know! It was so satisfying!!!

[@MurakDurak]

integral of sprt(tanx)dx has to be my favourite integral, but differentiating it is even more crazy!

[@shiina_mahiru_9067]

at 10:24, instead of changing tan x in terms of sin x and cos x, I would just multiply sqrt (tan x) both upstair and downstair to the second fraction. That should be the first thought because you want to end up with sqrt (tan x) after all the simplification, and to make it happen, sqrt (tan x) must be in the numerator

[@MrPhilipp456]

This is insanity! I love it

[@JulianOnions]

This is amazing!

[@bsul03420]

Excellent, but now I have to differentiate my inverse coth expression to show it's the same!

[@berenjervin]

Great video. Makes me want to go back to uni for a math degree after I retire!

[@MarkMcDaniel]

Brutal, but satisfying as hell.

[@cheeragsridhar1696]

It’s so awesome The rate of terms falling from my eye is differential

[@BRUBRUETNONO]

Ace derivative master ;)

[@IISH4RKZzII]

This is SOOO DOPE !!!

[@axel15473]

He is so on fire XD

[@helved807]

this is the most satisfying differentiation

[@TheYoshi463]

Yay, blackpenbluepenredpen made a new video!

[@blackpenredpen]

I did yesterday too

[@Lily-zd6dx]

Mindblow

[@johnsalkeld1088]

At the stage of getuve rid of tan plus cot recall that this is cot * [1+tan^2] so cot * sec^2 easier I think due to direct cancelation and resolving the full Eqn to 1/(cot * tan^1/2 ) or tan ^1/2. but from a pedagogical standpoint I see that using a standard method is good

[@wydadiyoun]

How about starting from square tangent and find that expression by integrating

[@July__-eu5pd]

Wow you actually differentiated it :p

[@XXgamemaster]

Beautiful.

[@dyer308]

Please show us Feynmans trick on integral solution of sinx/x without a power series solution!!!!!!

[@calcul8er205]

raees khan Another cool way that uses the idea of multiplying the integrand by e^(-st) is by double integral and Laplace transform

[@garomo6384]

amazing !!!!

[@gvchas]

Everything is possible. Just believe in your math!

[@bhavydugar6665]

This is just so coooool I took its SS and made it my wallpaper

[@Justiin_rm]

Woaaaa

[@eetuhaavisto524]

Hey blackpenredpen can you solve this derivative next time with the definition of derivative? That ought to be an interesting limit(needless to say that very fun as well)😂

[@blackpenredpen]

I prob. will have to use 4 classrooms for that...

[@fakeplaystore7991]

This is evil sorcery, pure and simple.

[@osvaldomena1172]

Amazing man

[@blackpenredpen]

Thanks!!!!

[@TheScienceGuy10]

Could you please do a video of integration of sqrt tanx? The other way round. I've been trying it out but getting stuck. Would really appreciate it.

[@jmannthemann4793]

Siddhartha Pundit He already has, just look for it

[@id120266737]

You're a fucking genius.

[@user-gt1cn2tf7s]

Miraculo!

[@AdrianMartinez-lt4dz]

Man! You're a genius 👌 Now do it in chinese xD

[@barbarodave]

Why?

[@anthonysteinerv]

I used trigonometric substitution for both of the integrals and got the exact same inverse tanget but I got a logarithm, the thing is that I did the derivative of that anseer and I didn't get the square root of tangent x, actually I got the sine of 2x, I need help bro.

[@vibhupandya6103]

Hey in sqrt cotxtanx won't it be = sqrt1=±1???

[@elementalsheep2672]

from about 12:00 onwards you could hear him getting more and more excited

[@chessandmathguy]

beautiful. just beautiful.

[@wduandy]

Love you

[@ck7671]

8:19 simplify the 2/sqrt(2)^2 !!

[@PackSciences]

Valid forall x in [0;pi/2[ mod 2pi. You could have made it more impressive using the formula arctan(x) + arctan(1/x) = sign(x) * pi/2 in order to get an uglier expression.

[@blackpenredpen]

LOL!!! maybe next time

[@axel15473]

Oooohhhhh yessss. I wait for that. The other way

[@Xeotroid]

1a) Integrate sqrt(tan x). 1b) Check the answer of 1a by differentiation. 1c) Prove 1b.

[@obinnanwakwue5735]

Let epsilon > 0...

[@alomirk2812]

Just wanted to point, this is the solution of the integral but it is undefined in the reals, if you will extend it to the complex numbers it will be defined and give you a real number

[@imcalculus]

Do this with definition of a derivative

[@ReubenMason99]

Could you possibly do some more detailed explained integration videos for people who don't know much calculus

[@francoisgatine4106]

There is a youtube channel called Professor Leonard (hope I wrote it correctly) who records himself teaching several maths classes. He's a really good teacher in my opinion, it can help you with understanding calculus. No, it isn't boring.

[@blackpenredpen]

Reuben Mason hi. U can also check out my playlists. www.blackpenredpen.com/math/Calculus.html

[@kindlin]

That's a detailed list. How did you have the time to put all that together with all your regular classes/study etc.?

[@wellbangok8959]

I'm a simple guy. blackpenredpen uploads a video, I watch.

[@blackpenredpen]

Sheev Palpatine lolll thank you!!

[@golocok]

Un final de cálculo que te haga resolver la integral de la raíz cuadrada de la tangente y luego derivar para demostrar que el resultado estaba bien. No aprueba nadie. :D

[@afsarshaikh9654]

Mind Fucking Blown. Just see how beautiful that is!

[@AnOrrangeShirt]

Would it be possible to integrate sqrt(tan(x)) twice?

[@hungryfareasternslav1823]

Yes, but that will be real pain or it is non-elementary!!!

[@ToeNailMuncher111]

How does he do the chain rule so fast

[@yopxy6188]

Do it with limits

[@ExiDJuicy]

Missing "+C" at the beginning 😎

[@OonHan]

DIFFERENTIATION DOESN"T NEED A +C

[@unknownanonymous5914]

+Oon Han thejoke.png

[@arekkrolak6320]

If you have to remember derivative of arc tanh then you can also remember what id the final result :) be a real man and use definition of derivative! ;)

[@mbrq1196]

Waw

[@majkgmajkg2613]

I'm sorry early too... In my country is 0:14 am. On 7 I have job... Shit...

[@yahyabajdadi516]

How about differentiating sqrt^3(tanx)?

[@natan9065]

Yahya Bajdadi d/dx((tanx)^(3/2)) = (2/3)(tanx)^(1/2)•(secx)^2 By the chain rule

[@azzteke]

Almost nowhere in the world are the sec and cosec still taught.

[@lalitverma5818]

I can't imagine

[@upanshulakhani6221]

That's nice but why don't you check the answer by integration

[@200Exotic-Topic]

Integrate your answer again to double check

[@colleen9493]

Now to the integral of the integral of sqrt(tan x)

[@stolenmonkey7477]

I kinda just went along with the ride lol, wow

[@upanshulakhani6221]

Now check the answer by integrating

[@maxhaibara8828]

what's the derivative/integral of N! (factorial of N) ?

[@MagicGonads]

No idea, but I found interesting that n! = int(0->+inf) (x^n e^-x dx) for x > -1 (Found that by accident) also written as int(0->+inf) (e^(n*ln(x)-x)dx)

[@maxhaibara8828]

wow he need to make a video about this

[@MagicGonads]

It's not a derivative because it has both n and x, but still suprising.

[@MagicGonads]

www.desmos.com/calculator/oukquiyntu

[@MagicGonads]

www.desmos.com/calculator/balkjdohcq The curves are calculated, not derived.

[@gloystar]

What a mess LOL!

[@agustindominguezcruz5000]

Now do it by the derivative definition

[@Wu-Li]

int of √tanx dx tanx=u² 2udu=sec²xdx dx=2udu/(u²-1) again, int of √tanx dx = int of u×2udu/(u²-1) = 2[int of u²du/(u²-1)] =2[int of (u²-1+1)du/(u²-1)] =2[int of du + int of du/(u²-1)] =2[u+int of du/(u²-1)] =2[u+½log|(u-1)/(u+1)|] +k =2u+log|(u-1)/(u+1) +k =2√tanx + log|(√tanx -1)/(√tanx +1)| +k Therefore, Int of √tanx dx = 2√tanx + log|(√tanx -1)/(tanx +1)| +k

[@OonHan]

covered my ears during the time of 0:15 The integral of *I CANT HEAR ANYTHING BLA BLA BLA*

[@blackpenredpen]

lol!!! Let me integrate blah blah blah next time!

[@OonHan]

LOL

[@pepemeade8845]

Holy jesus...

[@Bludgeoned2DEATH2]

Now do it in Chinese, Dr. BlackPenRedPen!!! 😁😁

[@blackpenredpen]

Will you watch, like, and share it?

[@Bludgeoned2DEATH2]

blackpenredpen of course, just like I watched the integral of sqrt(tanx) in Chinese and liked it too!

[@blackpenredpen]

Challenge accepted!!!

[@physicsphysics1956]

Where is the mistake?