Calculus 2 Lecture 6.3: Derivatives and Integrals of Exponential Functions

[@darthboganis1356]

euler's number 0:00 Definition of e 1:20 Properties of e 5:00 Algebraic examples 8:30 > recognizing a quadratic 18:30 Properties of f(x) = e^x graph 24:30 limit of a function involving e^x 34:00 Considerations when function is infinity/infinity 44:15 Calculus involving Exponentials 45:00 > derivatives [including d/dx e^x proof] 45:30 [Remember derivative of ln y involves using chain rule] >derivative e^f(x) 49:00 w/ Examples 50:00 More examples 1:05:00 Integrals involving e^x 1:07:00 > Integral of 1/u 1:21:15 > Integral of sec (u) 1:26:30

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thanks!

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thank you so much

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ALWAYS WATCH EVERYTHING IN 2X SPEED, unless you don't understand the basics. But if you don't know the basics, don't fret because Prof Leonard has us covered with almost everything. Somebody please give this man an award to honor his greatness.

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Leslie Anne Mujica it is easier to take notes at original speed

[@user-yr3uj6go8i]

Yeah, I'm considering doing that. The videos are way too long for me.

[@radhikarajput1984]

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2x too fast, 1.5x is the best with several pauses to take notes

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The 45 min are perfect lenght

[@vhp310]

Taking a 6 week Calc 2 class this semester and Professor Leonard is saving my life. Thank you sir, you're an American hero

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How did you do?

[@HeavyReign12]

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[@vhp310]

@@yoyonostring2147 got an A

[@vhp310]

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[@yoyonostring2147]

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Professor Leonard is one of the few professors that can actually teach in a way that people can understand.

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Saved my time and money, and by extension my life. Thank You.

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[@Abdulhaleem_6]

Real.

[@silverweeaboo1797]

Loves how he makes sure you get the algebra down. My calc teacher expected to know algebra that I haven't taken in 3 years, and also did not do any algebra prerequisite classes. So i was confused by algebra, thanks for explaining

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I FEEL U

[@radhikarajput1984]

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[@dingleberry869]

Season 2 of calculus is pretty good so far, but I miss that one kid from season 1 laughing at everything. Some of the plot can be hard to follow really hard to follow and I feel like a real nerd when I have to write some of these events down on paper to keep them straight in my head. I've already filled 2 and a half notebooks.

[@saadrehman362]

how'd you fill over two books by watching 3 videos?

[@ksrajavel]

@@saadrehman362 He must have meant the calc 1 inclusive notes

[@alenaalena9322]

Yeahh that one kid

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Bruh have u watched season 3

[@emiandere2560]

Yeah, I miss Scotty too

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What a great instructor, I already knew the best math teacher in the Middle East and Balkans but I have learned the best of US as well. Beton yetmez hoca, adamsın

[@OwolabbyAzeez]

God bless Professor Leonard

[@terahnewbold6124]

Eureka .....Professor Leonard you are the College student's lifeline....Thanks sooooo much!

[@King_Kato]

Professor Leonard is literally the G.O.A.T. I wish I had the opportunity to have him teach me at a university. Keep being awesome man! I'm starting Calc 2 and I would much rather watch these videos than watch my professors videos.

[@radhikarajput1984]

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[@azreow]

I'm in Calc 2 for my second time and for the first time since I've begun the class I was finally able to answer a question without looking at the work on the internet first. Thank you so so so so so so much!

[@radhikarajput1984]

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[@Robot_Chimp]

Did you use him for the first time you failed ?

[@azreow]

@@Robot_Chimp Wow, strange comment to reply to a 5 year old comment. The answer is obvious, no. The calculus teachers I had lectured on proofs only and worked 0 examples.

[@gangsta412]

​@@azreowrespect for answering after all these years haha

[@azreow]

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[@Morax___]

this is like quadrillion times better than what I paid for

[@fyijamy1534]

Professor Leonard, thank you so much for helping so many students. Do you teach Calculus 3? I will be taking calculus 3 next semester and I want to prepare before the school begins. If you do teach calculus 3 then please please please film calculus 3 lectures and upload them. Thank you again.

[@possiblepilotdeviation5791]

I've got Cal II coming up in a few weeks when the Spring semester starts, and I've going to kick its *** with the help Professor Leonard! Thanks so much.

[@samueldarenskiy6893]

Lyle Arnett, Jr how did it go?

[@VincesArtDesigns]

How did it go????

[@melanieelise3283]

Calculus being so tough for me brought me here, thank goodness, and currently in the middle of a pandemic!

[@TrungNguyen-xj7zx]

Thank you so much Professor, with all my respect, you saved my cal life. My professor goes so fast and I couldn't follow him. Your lectures so amazing and so easy to understand. Thank you

[@radhikarajput1984]

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[@ghostfrom1950]

man, my interest in math has been fading with the advent of remote learning of the pandemic. I would love to have a teacher like this during these times

[@joonlee5626]

Thanks Boss.

[@andilemdlalose9926]

the guy is the best.............

[@williamfunderburk7317]

Dont know what i would do with out you youtube vids. probably have to go to the math lab. When i cant remember what was lectured about the previous week I just pull up your channel. Youre So helpful and thorough. Thanks

[@audreyandremington5265]

Where's my "Why are we learning this in Calc 1?!??!" crew at?

[@snafu4714]

Bro for real wtf are we gonna learn in calc 2 after this

[@audreyandremington5265]

@@snafu4714 how to find the surface area of a squiggly line that's rotated around the x axis, how to break integrals like xsinx up into two different parts, and more fun stuff. Basically, I'm taking calc 2 rn. It's hard.

[@saymyname5942]

kinda late, but this is calc 1 review

[@radhikarajput1984]

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[@radhikarajput1984]

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[@techjesus99]

I will pray that my future children have teachers like you.

[@zoommier8220]

1:21:12 "Look at that" I love the genuine awestruck reaction of one of the students from seeing the simplified integral

[@xrisku]

i did the problem at 1:06:40 and got it correct. looks like watching these videos is paying off. I'm self-teaching/not in any formal class, though these are like a class. I was beginning to think I'd never catch on. yahoo! thank you!

[@2kchallengewith4video]

I need an update

[@tylersteffensen9268]

"Let's all try to get it right this time, 7 or 12?" ((class gives various answers)) "Oh dear lord."

[@ToriFurlotte]

god bless I'm so happy i found this. I'm in calc 1 and am learning this for some reason and was lost when your calc 1 lecture 3.1 was not this haha, thank you, you save me!

[@JP-sw5yp]

Sitting in a starbucks doing this, and when you proved that d/dx[e^x] is e^x I actually said 'oh my god' out loud and people looked over xD Professor, thanks for making math fun! I'm a math major anyway, but after watching you I'm even more pumped for what's to come!

[@ProfessorLeonard]

This made me smile :) Thanks for sharing that Jacob!!

[@JackDoesGames]

Found this guy on google looking for stuff to learn about Calc 2. Taking Calc 1 right now and it's moving a bit slow for me so I found him and it's enjoyable to learn ahead.

[@pinkisharma9876]

Professor Leonard sir the graph you have drawn at 26:30 is actually the graph of f(x) = a^x where a > 0 and not e^x since 0 < e < 1, e^x will keep DECREASING as x approaches +ve infinity and not increasing as any no. less than 1 will decrease with increase in power. I hope u take a note of it sir, keep doing the hard work you're a superman :)

[@odayadel3605]

e is 2.71 therefore > 1 so no

[@anonymoususer4356]

Best Math Professor of all time

[@daniellin3279]

thank you so much Professor Leonard, I'm watching your videos to ace Calc 2 in this summer.

[@isaacfajardo8627]

Prof. thank you for teaching me not just how to solve but you also teaching me the concept itself. Many students know how to solve complex math but they do not know where it came from or how to use it in real life.

[@radhikarajput1984]

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[@Litt13F00t]

21:08 that long streak of quality puns was spectacular. Good content. Good prof. 10/10

[@aarond.8656]

First time watching a video, and I only watched the pieces i missed from my own lecture- but definitely subscribing for ( 39:04 ): Pr. Leonard: How much is 1? Class: 1 Pr. Leonard: :-) that's nice, I like that.

[@sanjaygoyal5395]

this superman is a god of maths

[@abdullahmohammad5613]

That moment when he was asking about if they remember the exponent rules !!! LOL

[@andygallegos7212]

Professor Leonard is the GOAT!!!

[@ShivamYadav-ys3bp]

Thank you professor👨‍🏫 thanks 🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🌹🌹🌹🌹🌹🌹🌹🌹🌹❤❤❤❤🌹

[@Primitive_Code]

He looks like the guy or professor you see in LMN movies that seems so calm and respectful but if you upset him, he'll destroy you in a heartbeat.

[@DroughtBee]

Just a small tidbit of information at the very beginning, it’s generally agreed that Euler didn’t name e after himself and it just came up as a variable as if he did want to name it after himself he would have used a capital E.

[@Joabfilandus001]

Example 6 , was in my end of semester test ❤

[@WhatTechShow]

I learned all of this in cal 1... WTF I HATE MY SCHOOL. Almost killed myself over calculus!!!!

[@awesomewinter3103]

I feel ya. My professors have put calc 1 and 2 into 1 semester. It's a stupid thing to do really. In the end all you learn is plain calculating without any understanding of what you're actually doing.

[@kharnakcrux2650]

+WhatTechShow pshh... it's baby stuff

[@awesomewinter3103]

kharnak crux Never overestimate the self evidence of science.

[@BenBowman97]

That sounds awful. How did you survive?

[@Peter_1986]

@@kharnakcrux2650 Eh whatever dude, Arithmetic is even more "baby stuff", yet it is the most important math subject that anyone will ever learn. You would literally not stand a chance in any future math course if you didn't tackle Arithmetic first. And yes, Arithmetic is part of math - that's what you making use of in literally every single math or science equation in history.

[@georgesadler7830]

Professor Leonard, thank you for an exceptional video/lecture on the Derivatives and Integrals of Exponential Functions in Calculus Two. With constant practice/repetition Derivatives and Integrals are simple to understand in all levels of Advanced Mathematics. Professor Leonard at the 55:40 minute mark, when you differentiate (x^2 e^(-2x)), your solution is (2x e^(-2x) - 2x^2e^(-2x)), which is incorrect. The correct solution is 2xe^(-2x) - 2x^3e^(-2x)). Please correct this error in the video.

[@hollywood4166]

where did you get the -2x^3 from?

[@jacobstreet9058]

Thank God for this during Coronavirus!!!

[@andrewarcher1983]

am soo glad i found you very early

[@maksymandrzejdrzyzgiewicz5096]

19:50 Man, I'm not gonna lie. I've been self-studying with Leonard for a year now and went through 3 1000 page books with him doing everything from pre-algebra all the way to here. Doing everything alone is difficult in a way that you don't really have a frame of reference like you would in a class. In a class, you generally know how well you're doing because 1: you're getting graded and 2: you see how the other students and the teacher are progressing. I've been struggling recently because I'm not entirely sure if I'm doing well. The moments like these, where it appears that everyone isn't a mastermind and that I'm actually doing well, ah, magish.

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Thank you Professor Leonard. You have made me to understand a number of topics that were initially difficult to understand. God bless you.

[@radhikarajput1984]

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[@ВадимШатов-з2й]

it should be on netflix

[@rul1175]

Awesome videos. I believe the ideal lenght is 40 min.

[@shahzadhasan8369]

great lecture. well explained. thank you

[@Digitalsmilez]

Thank you so much. Wayyyy better than my UCD professor

[@MaryumNoorX]

you're my only hope in calculus.

[@louisramirez975]

Professor Leonard is my professor, after I watch his vids I go attack mode on the homework lol

[@fatimazaghab5855]

Lots of thanks Prof

[@ghosteffy]

Something get this man a goddamn medal

[@tanviboppudi724]

It'd be great if you posted videos of multidimensional calc too!!

[@tedchirvasiu]

1:09:00 You could split e^2x into e^x * e^x and make one u and the other du, so you get Integral(u du) = (u^2)/2 and when you substitute back it gives 1/2 * e^2x + c, right?

[@charmendro]

Another awesome Video!!!!!!

[@CurbEnthusiasm]

When your own professor tells you that infinity/infinity=1. Thank you Prof. Leonard, I'm adding your knowledge to my pokédex. :P

[@luke3319]

Did you skip a step in the problem at ~40min? You say you're dividing by e^-2x, but isn't the idea to divide by an expression that evaluates to 1? Like (e^-2x)/(e^-2x)? It looks like that's what you did in your algebra, but it doesn't match what you said and wrote on the board. Dividing by just e^-2x wouldn't impact both the numerator and the denominator

[@motivatedmind11]

Hello hot and good at lecturing math teacher lol very helpful video!!

[@karamarachid1886]

AWESOME EXPLANATION. THANK YOU SO MUCH

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thanks your videos really helps

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thank you so much!

[@SteamPunkLV]

21:08 I rewatched it and I actually understood it xD

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Great teacher.

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Thank you you made my dream come true

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Yes need more tricky treats!!

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I feel confident.

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GOAT!

[@ceeq]

اروح فدوة لك لحد يلبس اسود بعدك

[@maksymandrzejdrzyzgiewicz5096]

1:13:15 'Yeya'

[@elifozdogan9003]

nice !!!

[@adrianloh6919]

@ 1:28:48, I thought the argument (e^x + 1) NOT SAME as the denominator (e^x + 1), but why they mentioned ln u/ u, as u=(e^x + 1) ....so confusing

[@fatimadarling9656]

I’m taking this in calculus 1.

[@michaelzanfardino5931]

the example at around 32:50 since the limit from the right and left are different than the limit DNE?

[@alondras8227]

On 1:26:37 how did he get the ln? I thought we only use ln when it's (1/x)?

[@gentlemandude1]

"Don't forget those freaking parentheses!!" (9:45) Okay, okay, I won't.

[@luiishu535]

At 1:12:18 I picked e^(2/x) as my substitution and got the following solution: ∫[e^(2/x)/x^2]dx u = e^(2/x) = e^(2x^-1) du = e^(2/x) * (-2x^-2) * dx du = -2e^(2/x)/x^2) * dx dx = du * x^2/(-2e^(2/x) ∫[e^(2/x)/x^2]du * x^2/(-2e^(2/x) ∫[1/1]du * (-1/2) ∫[(-1/2)]du = (-1/2)u + C = (-1/2)e^(2/x) + C What confuses me is that I got the right answer. Did I do anything wrong, or is this solution acceptable?

[@christopherrodriguez1891]

Thanks for the initial review very helpful!

[@Diverse27]

@1:11:30 -- Don't we integrate by adding 1 to the exponent and dividing by the same number? with regards to U-substitution, if U=2x, then shouldn't S(e^u) result in e^u+1/u+1 = e^2x+1/2x+1? Sorry if this is off base..

[@ziggaby]

1:17:30 "We're gonna have to attack it directly. Now, the only way we can attack this thing is with a substitution." Starts to sound like a weird video game.

[@anshumanagrawal346]

How do people get through Calc 1 without knowing the exponential function? That sounds impossible

[@emadahmad605]

My Teacher, Professor Leonard, thank you, Indeed I have some issues that I did not understand how to solve. Can you help me.

[@frenchmike]

lne = 1. yeah, but log 10 = 1, log 4^4=1. any log with same base and x = 1. so how can this make e special

[@nintran8861]

my last probem's aswer: (ln(e^x+1))^2/2 + C

[@samueldarenskiy6893]

This is a great video, but didn't the students learn this in calc I

[@farhanahmed7729]

which book you are following?

[@bobmpotwane5338]

You should come and lecture at university of Johannesburg 👍

[@RaiSarfrazHussain]

How we came to know that we are studying which calculus? Like is it 1, 2 or 3?

[@ryanmetevier9141]

If my classroom was this quiet when the professor was teaching maybe I wouldn’t need to watch this so bad 😂 or maybe my teacher is just shit

[@studyhard128]

The joke at the 21-minute mark. 🤣

[@chriss6439]

In the last example, why isnt the integral -[sec(e^-x) * tan(e^-x)] + C . Where did that ln[ ] come from can someone help me ?

[@madlyy187]

do you have a video for cylindrical shells or finding the volume of a solid obtained when a region is rotated about an axis?

[@tunminnnn4638]

Check out his lecture: Calculus 1 Lecture 5.3 in the calculus 1 playlist.

[@kepperan]

Anyone else notice that he got married since last season?

[@zzzoldik8749]

prof can you make video about how to solve integral e^(1/x) dx please... because its for my last task / scrip / essay

[@LostArkLover]

I love his reactive attitude when the students get it wrong! LMAO