Calculus 2 Lecture 6.4: Derivatives and Integrals of General Exponential Functions

[@barryhughes9764]

The students in this class have no idea how lucky they are, his enthusiasm for his subject is infectious, combined with ability to punctuate his explanations with witty remarks....he is quite remarkable. There are those who know what they teach, and there's those who 'can teach' what they know, and there is a world of difference between the two. Professor Leonard is of the latter, much rarer breed. Long may he reign.....supreme. He truly deserves all the success in the world.

[@francoisegiraldo5038]

You're the reason I was able to get my engineering degree🙏🙏🙏🙏 thank you Professor !

[@gabrielastankiewicz6361]

That's someone who has the skills to be a professor .. he can make Cal 2 seem like a piece of cake..... and fun...

[@shadmo8629]

Hello Professor Leonard. Thankyou for sharing your wonderful series of lectures. I'm a Calculus teacher and I often watch your lectures before I teach new topics. Yes, these lectures are great for teachers too. I've spent many hours watching you and patrickjmt. :)

[@whitb62]

I had an awesome Calculus 1 professor but something I really like about Professor Leonard is the he does the proofs before giving you a "rule/formula." I think this is incredibly important for really learning math.

[@dcfilms4425]

Best f**k'n CALC teacher ever. I've been through college and now university in Canada paying lots in tuition, and i have never gotten derivatives and integrals like i do now. Keep it up Professor Leonard.

me too, so sad...

[@lightning8542]

Best Calculus professor on the internet. Thank you for hard work.

[@leslieannemujica9912]

The only professor that I have ever "had" that actually teaches properly and still has time to incorporate proofs. You should teach at every university! They should just replace lectures with you and your videos, because nobody ever learns anything in those overfilled-sleeping halls.

[@iPoopRays]

I'm watching this during lecture instead of paying attention to my actual professor

[@Cody8P]

i walked out of class because my teacher does not do anything on the board and just always says "you should know this" watching his videos shows me even more how much attention this stuff needs

[@xgerbil3586]

I hated when professors used to tell me this. I'm asking a question because I don't understand. "You should know this" isn't exactly helpful.

[@sampuckett9082]

I thought the exact same thing today in lecture! I didn't end up going through with it because I figured that if I sat through lecture and then watched Professor Leonard's videos I'd understand them even more!

[@karimkarimli9614]

When he said don't pack yet, i felt the joy of packing that I used to feel in good old days.

[@faiqfarhan2309]

Im following Your Lecture Since calculus 1.Since Then My Marks Has Improved Alot.Now Im Study Calculus 2. Tq So Much Prof!

[@adeldalgamouni3933]

lmao who else misses the laugh of the guy from calculus I

[@dianampm99]

@John Napier i wonder whats hes doing. Id never stop going to calculus class since this man can´t be found anywhere else. Id be a student forever

[@feilongish]

5:00

[@igneouswatchman]

Bingewatching your videos in 2018; Professor Leonard is tried and true!

[@AlaaAAnani]

Thank you so much Professor Leonard! You are awesome.

[@lynnkang2827]

Thank you very much Professor for putting this lecture up online. I learn a lot from you because you explain very well. Please post some more lecture for Calculus

[@elifozdogan9003]

Great !!!!! Thank you so much!! It is the amazing class to learn myself these chapters during the summer!

[@cyrillemagdi7717]

thank you, I really wish all calculus teachers explained calculus as you explain it, keep up your amazing work

[@imamalam4971]

this prof is awesome !!

[@zerocks2294]

x^x actually also just works if you rewrite it as e^xln(x) and then just take the derivative with the chainrule

[@kkamous7278]

My respected sir,,,, i am a student of BS mathemathics ,,,, i lesson your lecture very attentiveness,, you are such a great man in matemathics,,,, i request to you plz make a "TTP " on the following question,,,,,, 1) why we take ''sin'' as per/ hyp,,,,, why we can't take hyp/per or base/per ........ etc, also we do same case for other trignometric ratios,,,, 2)from where the values of "log" and "ln" come from,,,,, i have weak english but hope you will understand,,,,, thanks you for giving me a time,,,,

[@trush1090]

Left side be slacking

[@henryponnefz1419]

Thank you for sharing your knowledge and wisdom. I am understanding things I never understood before.

[@s.o.k.1393]

Dude. If I pass my Calc 2 course it's because of you.

[@tampabaystormchaser]

Thank you very much for all your videos! I'm using your calculus 2 videos to help me with my class, and will be using your calculus 3 and differential equations videos to help me in upcoming semesters!

[@rileynobles7146]

You are an amazing teacher!!

[@Gozi101]

Duude your "can't touch this jokes" XD love your teaching. keep on the good work :D

[@carlosg4575]

Thank you Professor Leonard. Trying to get ahead for Cal 2 next semester. Glad I started early as it seems alot will be going on.

[@adriangil1277]

I'm taking calc 2 and my professor is so slow that we reviewed derivatives, integration, and logarithmic the first week. The class is 8weeks long and I'm learning from professor Leonard and I'm killing it on the homework!

[@nemuirostorageroom]

my prof on the other hand, it's still in the middle of the first semester and she's already covering half of Calc 2 already.. we're supposed to be learning calc 1..

[@m7mdarwani964]

"Don't take calculus to show me that you have failed algebra" Professor Leonard.

[@ShivamYadav-ys3bp]

Pranam gurudev thanks 🌹🌹🌹🌹🌹🌹🌹🌹🌹🌹🌹🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏

[@johntindell9591]

Thank you very much, very helpful indeed

[@jotlewis368]

8 years later and I'm watching all your videos to review for my exam tomorrow. Hope no rules have changed lol

[@jkgan4952]

lollll im watching and i'm doing calc 2 next school year

[@jkgan4952]

hope exam went wekk

[@jotlewis368]

@@jkgan4952 it did not but that’s because I tried to cram instead of studying the week before. This guys channel was seriously perfect help though highly encourage you keep using him

[@jotlewis368]

@@jkgan4952 calc 2 is a bitch it moves fast but as long as you got all your basics down to a science it’ll be easy

[@jkgan4952]

@@jotlewis368 OK cool thank you! Sorry about the exam. U got the next one!

[@evanhumphry9263]

The guy closest to the camera is great. x^e is ex^e-1, "Whaaaa?" made me laugh.

[@dhuratabali11]

Yes , because e=3.14

[@pandabreaddoesmath12]

@@dhuratabali11 e is 2.71, pi is 3.1415

[@rumpelforeskin8461]

"Whoever's phone that is, turn it off now" said no teacher ever

[@mail2nadapurayil]

Kai Evans he said that so nicely!! In my college, professors will come and take it off and return it at end of class!! Prof Leonard did it so good without hurting his student feeling, he conveyed the message....

[@Peter_1986]

@@mail2nadapurayil He also doesn't put students on the spot when he asks them questions, and instead just says "SOMEONE answer this, please". I always hated when my teachers told _me_ to answer some of their random questions, and then I would feel stupid if I didn't know the answer.

[@maximilianbur2560]

@@Peter_1986 if you don't want to feel stupid why don't you just know the answer? saves me all the time

[@Peter_1986]

@@maximilianbur2560 You don't always know the answer to every question that you are asked. And it's really irritating when a teacher puts a student on the spot like that, so it's obviously much better to invite EVERYONE to answer the questions, so that those who actually know the answer will give the correct answer immediately.

[@sardarbekomurbekov1030]

Excellent professor, thank you!

[@dogblues4829]

Great

[@joshuagaa7009]

Nice nice

[@paulschuebel5487]

The disappointment in his eyes at 35:54 when everyone said, "the 'y'"

[@samueldarenskiy6893]

When professors in community college are 100x better than the ones in private and state schools.

[@dhuratabali11]

You are fantastic

[@georgesadler7830]

Professor Leonard, thank you for an excellent explanation and analysis of Derivatives and Integrals of General Exponential Functions in Calculus. This topic in Calculus shows that any function can be differentiated, however not all functions can be integrated in closed form. Professor Leonard, at the 1:07:00 minute mark, you have d/dx ( log base a IxI) = ln IxI /lna, which is not correct. The correct solution is d/dx (log base a IxI) = d/dx ( lnIxI/ln a). Please correct this error in the video.

[@aroundtheworld52]

Professor Leonard, Do you have copies of the integration tables and trig identity tables that you reference in your videos? I am trying to learn Calculus in a short period of time, and I think those would help a lot. Please let me know Thanks!

[@gabrielbartels5879]

why can't all professors be like this ?

[@kylewanamaker9109]

Leonard: Whats the first thing you notice about the problem that has to be done? Dude that went to the wrong class: There's an equal sign

[@Peter_1986]

I prefer to think of calculus as "students take calculus to finally NAIL algebra".

[@philopoemen6659]

28:51 y = x^x, for me it's simpler to take y = e^xlnx, and take the derivative of that. So dy/dx = e^xlnx (lnx + x/x) = e^xlnx (lnx + 1) = e^xlnx * lnx + e^xlnx = (x^x)lnx + x^x.

[@georgekhalil4481]

Hat down professor thank you so much ❤

[@jakemcvey2926]

at 1:04:32 as written it seems like you are giving the answer to the the derivative but are you actually just rewriting the log? i.e. should there be a ddx around the ln|x|/ln(a)?

[@abdullahmohammad5613]

I just can't stop watching your videos :)

[@anupammishra5687]

Me too😯

[@eslamsamir3736]

1:07:37 why the Derivatives of d/dx log (a) (u) not =0 the function deals with u not x???

[@Kyrowebb]

Is the answer for d/dx[x(5^3x)] = (5^3x)+3x(ln5)*5^3x?

[@markjerusalem6944]

is this outdated? Like is the curriculum for calc 2 not following the playlist anymore?

[@ShivamYadav-ys3bp]

Thank you sir 🙏🙏🙏🙏🙏

[@LearnWithFardin]

Superman!

My teacher and everyone in my class, for some weird reason, use log and ln interchangeably. This is mathematically wrong! Thank you so much for pointing out the difference and how simple it actually is to understand it. I am stopping using log and now going to use ln, which is more appropriate.

[@G3gayathri]

❤thankyou so much sir

[@chase.7859]

What exactly does taking e=e^lnx mean at 3:00? What does it represent?

[@LUCATRON-hs8cj]

20:24 "And that's a good place to stop"

[@sdparsons]

When the guy at 28:00 is like "ohhhh do we have to factorise tho" (o.o) , and Pro Leonard is like yep, yep you do, made me laugh

[@brandonfell9871]

thanks so much, do you teach calc 3 those videos would be awesome

[@fazochgreatness7330]

24:22 Dziekuje

[@hassanlaghbi7047]

very nice

[@shutnawi6635]

thank you

[@peacemaker23134]

55:46, you forgot to substitute sqr root of x back for u

[@PelayoGilberto]

thank you so much!

[@danielglorioso9066]

I assume the ln(3) comes in after we do the interval. Thanks for these professor your t-shirt game is strong

[@danielglorioso9066]

because you did not solve for dx you just put all x's on one side and the u's on the other?

[@chase.7859]

I wish I could find someone who would explain --graphically or mathematically--what we are doing when we go between log and exponentials.

[@ullaullasen5871]

Check out professor Leonard's precalc playlist. There is a perfect explanation over multiple videos.

[@Squash101]

12:25 deravative of e^x proof

[@dougjohnson5487]

I am still confused why this logarithimic exponential method is used here instead of the regular differentiation method used in Calculus 1. When is one used versus the other? I don't think the prof ever contrasted the 2 methods. Can anyone explain? I checked an on line derivative calculator and that agrees with the answers shown here but not why the 2 methods.

[@ahamasarp6284]

hey,did you find why?

[@repellomuggletumify]

Can someone please tell me why there isn't a chain rule inside a chain rule at 42:17? I'm so confused. Like, the chain rule inside the ln comes to 2x+1 but why isn't there another chain rule which would result in 2?

[@thomasteal5609]

He told the class not to put 2x+1 and did the same thing himself. It should have been times 2. I got the general point though.

[@repellomuggletumify]

@@thomasteal5609 thank you!!

[@varshabhambhani4819]

could either of you pls help me understand where does 1/ln10 come from in last example i.e y=x^2[log (e^2x) +1]

[@abdullahmohammad5613]

The strangest thing is that I took alot of cal2 in cal1

[@GabrielaLopez-dc5wb]

in 55:59 where does the 2 in the u-sub come from? and why do we fully get rid of the bottom square root of x?

[@unknownperson7265]

We fully replaced dx/(x)^1/2 with 2du because they are equal, and we know their equality by rearranging the equation du/dx = (1/2)x^-1/2

[@unknownperson7265]

Everything is written on the board at 55:59, just look at it for a minute or two, or rewind it I guess

[@bryan_truong]

damn i wish he did a lecture on work

[@satheeshkumar3269]

1.05.10 ,Teacher says log a (x) = log (x)/log(a) .How does it work?

[@ProfessorLeonard]

This is the "Change of Base Formula" for logarithms.

[@satheeshkumar3269]

Thank you very much sir... I am honored to get a reply from you sir ....

[@mrmelanson34]

at 1:00:16, should his du be (3^x)/ln3?

[@richardpfeynman4526]

RyanM No, he is right

[@mrmelanson34]

thanks!

[@fake-chemistry]

nice black shirt professori

[@Euphemia-em8fe]

done

[@camillamunaco3753]

PROFessor .... PROphecy

[@camillamunaco3753]

lol the PROF never fails...may he reign!!

[@dhuratabali11]

I have one question d/dx(lny)=0 why you write 1/y ??????

[@dhuratabali11]

d/dy(ln y)= 1/y

[@benjaminjiang1333]

24:14

[@keaqan]

@37:34 when I tell my friends I have something very important to say then one of there phones goes off

[@moonquartzs]

Who else is here for their Calculus 1 class?

[@naruto247fighter]

47:16 that evil laugh do

[@user-eb9uj8zi6i]

37:30 :0

[@Mohammed-mm3kb]

1:02:40

[@MS-oy4vo]

thinking to myself... 'Holy shit. What's a derivative again?'

[@CrimsonHawk09]

I'm more confused as none of this sounds like what is being taught at lecture.

[@javierd2403]

CrimsonHawk09 really, because this is exactly what my professor teaches except she really goes in depth and provides proofs and all that goodie math sentence stuff. That's what confuses me and I come here to reinforce and in a way review what I learn or preview what I will learn

[@CrimsonHawk09]

Let me give a little background. 9th grade algebra teacher was a war vet who later on went to the crazy house after going off on god knows who. Before that event he would give 0's or 50's regardless if you were smart. God knows how the fuck I passed with a mediocre grade. On to tenth grade armed with nothing but my self taught Algebra. Geometry came and and passed with acceptable scores. Algebra II is next and I do quite alright (B+). Here comes precalc and a vietnamese lady. This is where it went down to hell. I couldn't understand more than 60 percent of her explanations made worse with her fast writing and only having 45 min per class. teacher was very nice and understanding but would get frustrated as I didnt understand. so this bad combination for education follows me to college trough trig, cal 1 and 2. Adding insult to injury my degree is computer engineering technology which involves shit loads of algebra, trig and calc.

[@asaleminik]

It's been 10 months, hope you found the answer bro

[@CrimsonHawk09]

+asaleminik already taking statistics

[@dovidbaum2229]

29:12 x^x

[@johnmayer1479]

" is there any kind of methodology that explains how to do each problem?" that would make me mad if I was him. That's what he's been explaining this entire time you dope.

[@johnmayer1479]

ask a more specific question if you're confused, as it stands you're just insinuating that he isn't teaching enough.

[@ouyangfeng6911]

15:29 crazy ! Many mistakes

[@sdparsons]

No mistakes here I think

[@varshabhambhani4819]

could either of you pls help me understand where does 1/ln10 come from in last example i.e y=x^2[log (e^2x) +1]

[@fatimazaghab5855]

Thank you so much Professor Leonard! You are awesome.