Seriously, I would fail calculus if it were not for these videos. I cannot tell you how large of a service you are doing the student community by posting these. You have my deepest gratitude. I'd be happy to donate resources for your services here!
In this example you have applied the chain rule once. In some tests it requires you to apply the chain rule until the inner function is at its simplest, so very basically you are left with one x and a number. That is why its called the chain rule: because you can continuously apply it until you reach 0.
@@kimmarch90 It's been a year but if it still confuses you, it makes more sense when you think of the exponent*thederivative has some constant (A) and apply the d/dx Af(x)=Ad/dx f(x)=A*f'(x) rule. Then repeat the chain rule.
+Khan Academy I just wanna say you are doing a marvellous job explaining the content of all your videos. You are the reason I could intricately understand certain concepts. Thanks a lot!! xoxo
I've been out of school for about 5 years now. Thinking about going back tor a mathematics degree I never finished as a Computer Science major. I have to say, Khan is more coherent than most of my university professors I've ever seen!!
These videos are amazing, I love how you are direct and straight to the point wasting no time at all. I am definitely recommending you to all of my friends :-)
i agree with all these people who find this video soooooooooo helpful....i was trying to study this in my calc1A text book and the longer i tried to understand the chain rule the more confused i got. this video makes it so clear. thanks a lot.
I'd give you 5 stars but I can only give you a thumb up. Thanks for explaining stuff in such a clear manner. You are going to be the sole reason I pass AP Calculus.
I appreciate the work done and time given in producing these videos but Good God - perhaps we could all chip in a few quid and enable Khan to purchase some more refined writing and video software. In other words, I suspect that many whom wish to learn Maths pause the video, in order to follow it through and reckon it in their own heads. But its almost impossible to see what is written.
Confused me? My teacher confuses me! This is what teaching should be, taking time to explain the steps. My teacher taught us the chain rule in one class (2 hours). The next day we were given an in class assignment with the chain rule, but with fractions and rationals. In my head, I was saying ''God help me''. Teaching is an art, and thank god there are still some good teachers out there sharing their knowledge for free. Thank you.
You didn't confuse me, very clear!! thank you. I am a maths tutor and I usual only tutor year 10 and bellow but I was recently given a year 12 student who "cries when it comes to methods, the whole class does" (His words!) this will greatly help me get him to understand this!!! so thank you again!! :D
i been struggling in calc this whole time n i tried my best to learn my text book n the more i tried the more it confused me but after watching your videos for d first time itself it all made sense...THANK YOU SO MUCH.. I DONT KNOW WHAT I WOULD HAVE DONE IF I DIDNT FIND YOUR VIDEOS. :)
i love you. u are so much better than my math prof who just doesnt even make sense. i think u are going to be the only reason y i pass my midterm tomro. THANKS. u are doing an amazing job. keep it up!
very good, i want to point out just one thing , in the last problem you need to put a parenthesis "or brackets"in [(15x^2-7)(5)(5x^3-7x)^4-6x^-3] so that way it would be [(15x^2-7)(5)(5x^3-7x)^4-6x^-3] (3)[3x^-2+(5x^3-7x)^5]^2
My god I feel so sorry for the people in the comments here that have teachers who don't explain it this way! My math teacher was excellent and she always made sure to go through everything step by step, even when we had two months less than everyone else to do calc (IB is hard). I'm glad these videos are here for people who aren't lucky enough to have teachers this amazing in school.
2:25 "To the minuuus 8." I don't know why, but something about the way you said that was hilarious. Thank you for making these videos and being awesome.
Excellent explanation and I learned a lot, so much that I could catch a mistake, somebody also did before me. Final answer: [(15x^2-7)(5)(5x^3-7x)^4-6x^-3] (3)[3x^-2+(5x^3-7x)^5]^2
I noticed you wrote a ^2 instead of a ^1, you were showing that there is a ^1 but when you copied it down in yellow you mistook it for a 2 :) Thanks for this anyway
wow. amazing. thankyou! my $4000 tuition money for the subject would be better off with you...atleast itd pay off by me actually learning something. thankyou soo much
This review well for me learning for others is great your sessions are well timed so never to long and easy to follow. I forgot how long these calculus problems can get. If I remember correctly some problems can get extremely long.
Omg... I actually got that.... if my University professor had attempted to teach me this then there is no way in hell i would have followed so clearly.
thank you so much for doing this! i can never learn properly in school because my mind wanders a lot, and if i miss even 1 thing the teacher says, i won't understand the entire lesson! but i can always replay videos and that's the best part
He does that, but in reverse order. Where the brown in the last line is the derivative of the large bracket, the pink times the green accounts for the derivative of the inner portion you mentioned: dy/dx (3x(-2)+(5x(3)-7x)(5). The light blue is the derivative of the last remaining inner portion: dy/dx (5x(3)-7x)(5). I was taught to tackle the outside first, so his method appeared a little backwards to me at first.
I like the videos, but I find it much easier to think about it in terms of a function of a function (of a function). eg. f(g(x)) or f(g(h(x))) For a simple one: dy/dx = d/dx f(g(x)) = f'(g(x)) * g'(x) For a more complex one: dy/dx = d/dx f(g(h(x))) = f'(g(h(x))) * g'(h(x)) * h'(x) For even more complex (like that example): dy/dx = d/dx f(x + g(h(x))) = f'(x + g(h(x))) * (dx/dx + g'(h(x)) * h'(x)) Maybe not the best notation to learn first, but makes solving ones like that last problem easier.
Howdy. I'm 13 years old. I started taking Trigonometry a month ago, and now I'm here. You must be thinking "Oh lord, you're humble aren't you?" But hold on. This comment isn't about me, this is about Sal. I had the worst teacher for Algebra 1 in 7th grade, so I went to Khan Academy. I'm 13, and I'm doing calculus. Thanks Sal.
i think it would make things easier if we used substitution,for example after taking the derivative of the inner function we can write on the side that it is equal to some variable, do all necessary steps then reintroduce everything at the end.
Ok, my only question is, in the second example, when deifferentiating the largst bracket, why wouldn't you then multiply by the derivative of the inside? so where you have 3(3x(-2)+(5x(3)-7x)(5))(2), why dont i need to then multiply by the derivative of inside the bracket?, so, 3(3x(-2)+(5x(3)-7x(5))(2) X (3x(-2)+(5x(3)-7x)(5)
Not gonna lie, when you said that that example at 8:30 was going to be one of the harder ones we'd see on our test, I freaked out--you obviously haven't met my teacher. I hope you'll have examples with trigonometric and/or logarithmic expressions!
Actually quackhail6 is not correct sorry. You can not distribute the -7 across the second expression because everything inside the second set of brackets is to the power of -8. To think of it another way, the second half could be written as 1/(x^3 + 2x^2 - x^-2)^8: to multiply it correctly, the -7 would multiply the 1 making it -7/(x^3 + 2x^2 - x^-2)^8. Hope that is clear, its difficult writing it out without algebraic symbols and proper "1 over"
discuss21 Sorry, drpbrabbit is right... remember PEMDAS. You cannot multiply -7 into the second term until you take care of the -8 exponent. The only way to bring the -7 into the parentheses is to raise IT to the -8th power first THEN multiply it. That would hardly be considered 'simplifying' so it is factored in to the first term instead. If you still don't see it, try it with an easier example: 2(x)^2 is the same as 2 (x^2) but is NOT the same as (2x)^2... it is actually (4x)^2. You have to square the 2 first, then you can multiply it into the parentheses.
Normal teachers: explains the simple questions and leave us with the hardest
Khan: explains the hardest and makes us feel that calculus is fun
But calculus is fun, indeed :/
@@r4fa3l59 these q isnt even that hard
Facts
This site has improved my grades immensely for several challenging college courses. Thank you.
B E I take calculus and vectors in high school
Sterling Archer cool no one cares
@@sterlingarcher3981 r/IAmVerySmart
@@JamesTownsendJian bet, I am an 11 y/o who learned this for fun without problem, redditor
The highest quality content available in the lowest quality
graphics!!
My prof is great but hearing someone else explain it is just that much more helpful.
I swear I will donate as soon as I can as you deserve it khan academy.
did you donate now ? after a year ?
@@deltaschool4302 All in due time.
Did you donate?
Did you😁
did you donate tho
Seriously, I would fail calculus if it were not for these videos. I cannot tell you how large of a service you are doing the student community by posting these. You have my deepest gratitude. I'd be happy to donate resources for your services here!
In this example you have applied the chain rule once. In some tests it requires you to apply the chain rule until the inner function is at its simplest, so very basically you are left with one x and a number. That is why its called the chain rule: because you can continuously apply it until you reach 0.
So basically is the derivative = derivative of the inner function * exponent! ?
I have been noticing this too. My professor takes steps so much further and expects us to do so also. :/
First order derivative. Only higher order derivatives require iteration of the chain rule.
@@kimmarch90 It's been a year but if it still confuses you, it makes more sense when you think of the exponent*thederivative has some constant (A) and apply the
d/dx Af(x)=Ad/dx f(x)=A*f'(x)
rule. Then repeat the chain rule.
Back in the days of 240p...
Back in the days of 1080p....
@@rayhanmansoor2951 though they have now come back!
To be honest it looks much more fun, entertaining, and captivating.
Good video but you should add natural logs, logs, trig fcns as well, that's where the chain rule raps a chain around your head.
The derivative of f(x) in the first example looks more complicated than f(x) to be honest...
im spending a fortune at university and learning 1/5th of the information i learned in this video
big ups :) and a BIG thank you!
Mr. Khan, you are a beast! The way you explain is amazing.
i have never had a teacher this great... bless you man, you're just amazing at what you do
You sir, are a hero above all heroes.
1925 views by today, in one year, this will have 1 million +
Thanks, Sal!!!!
+Khan Academy I just wanna say you are doing a marvellous job explaining the content of all your videos. You are the reason I could intricately understand certain concepts. Thanks a lot!! xoxo
Thank you always, Khan Academy! You are the best!
I've been out of school for about 5 years now. Thinking about going back tor a mathematics degree I never finished as a Computer Science major. I have to say, Khan is more coherent than most of my university professors I've ever seen!!
10mins and a chapter worth of doubts are now cleared!
I love you.....
i always use your videos to prepare for tests and exams. thanks for making math so much easier
still saving lives, i tip my hat off to you sir
Hey how's is it going pal
After 7 years
These videos are amazing, I love how you are direct and straight to the point wasting no time at all. I am definitely recommending you to all of my friends :-)
i agree with all these people who find this video soooooooooo helpful....i was trying to study this in my calc1A text book and the longer i tried to understand the chain rule the more confused i got. this video makes it so clear. thanks a lot.
I'd give you 5 stars but I can only give you a thumb up. Thanks for explaining stuff in such a clear manner. You are going to be the sole reason I pass AP Calculus.
I appreciate the work done and time given in producing these videos but Good God - perhaps we could all chip in a few quid and enable Khan to purchase some more refined writing and video software.
In other words, I suspect that many whom wish to learn Maths pause the video, in order to follow it through and reckon it in their own heads. But its almost impossible to see what is written.
this is from 2007 he prob upgraded
B J is correct. The video is from 2007. Not only was the technology not great at the time, video quality was also not HD.
This actually makes a lot more sense. My teacher taught us to go outside-in. This is a lot better in my opinion.
amazing... I watched these videos non-stop, and now I'm going to be ahead in calculus! (I'm in precalc)
Thanks so much!
How is your voice so soothing???
I have spent days trying to learn this and you taught it to me in 10 min. Awesome videos. Thanks for the help.
the world needs more people like you, sal.
Confused me? My teacher confuses me!
This is what teaching should be, taking time to explain the steps. My teacher taught us the chain rule in one class (2 hours). The next day we were given an in class assignment with the chain rule, but with fractions and rationals. In my head, I was saying ''God help me''.
Teaching is an art, and thank god there are still some good teachers out there sharing their knowledge for free. Thank you.
you are just changing the world man.That is something you should be proud of
A kid with rudimentary math skills can learn this!!!
Are you a some sort of math GOD?
I'm 16 and i understood this rule.
Thank you so much.
I watch your videos because they are entertaining.
These calculus videoes are top notch, better than education at universities and paid courses
You didn't confuse me, very clear!! thank you. I am a maths tutor and I usual only tutor year 10 and bellow but I was recently given a year 12 student who "cries when it comes to methods, the whole class does" (His words!) this will greatly help me get him to understand this!!! so thank you again!! :D
I got a 102% in pre-cal with your help, now its summer time and learning calculus too! Wish me luck on AP Calculus the coming year hehe :)
Never ever thought I would be smiling after learning Math.... You are math super fun... Loving it and getting confident with every passing video :-)
No you didn't confuse us. You r the one of rare best teachers globally. And i really like ur voice and the way you talk. Thanks sir.
i been struggling in calc this whole time n i tried my best to learn my text book n the more i tried the more it confused me but after watching your videos for d first time itself it all made sense...THANK YOU SO MUCH.. I DONT KNOW WHAT I WOULD HAVE DONE IF I DIDNT FIND YOUR VIDEOS. :)
My savior :'). I was scared, and instead of going to church, I went to Khan Academy.
i love you. u are so much better than my math prof who just doesnt even make sense. i think u are going to be the only reason y i pass my midterm tomro. THANKS. u are doing an amazing job. keep it up!
what a legend
dude u justify the quote that says " I bunk school to study in peace". thanks man . great help
I tried watching a different set of lectures first and they made next to no sense, but everything you say makes absolute sense! Thank you!
was waiting on that last set of brackets :) Great job as always.
Id been having trouble with calculus and its all gone after your tutorials. :) Thanks a ton, Sal!
dude you are the best math teacher ever man.... good job i get this stuff man thanks.!!!
very good, i want to point out just one thing , in the last problem you need to put a parenthesis "or brackets"in [(15x^2-7)(5)(5x^3-7x)^4-6x^-3] so that way it would be [(15x^2-7)(5)(5x^3-7x)^4-6x^-3] (3)[3x^-2+(5x^3-7x)^5]^2
My god I feel so sorry for the people in the comments here that have teachers who don't explain it this way! My math teacher was excellent and she always made sure to go through everything step by step, even when we had two months less than everyone else to do calc (IB is hard). I'm glad these videos are here for people who aren't lucky enough to have teachers this amazing in school.
It's about end of 2023 and i hope our dear teacher and all those who commented here are alive and happy 😊
sorry for my english 😅
im studying egineering and i have no idea what i'd do without these vids...
Khanacademy saved my life, period.
FOR THE MILLIONTH TIME, THANK YOU !!!!!!!! .
2:25 "To the minuuus 8." I don't know why, but something about the way you said that was hilarious. Thank you for making these videos and being awesome.
Excellent explanation and I learned a lot, so much that I could catch a mistake, somebody also did before me.
Final answer: [(15x^2-7)(5)(5x^3-7x)^4-6x^-3] (3)[3x^-2+(5x^3-7x)^5]^2
I noticed you wrote a ^2 instead of a ^1, you were showing that there is a ^1 but when you copied it down in yellow you mistook it for a 2 :) Thanks for this anyway
I want a shirt with Sal's face smiling inside of a heart.
And helping dreams too..I am back to maths after couple of years of break and here I found my class ! God bless you Sir !
wow. amazing. thankyou! my $4000 tuition money for the subject would be better off with you...atleast itd pay off by me actually learning something. thankyou soo much
You are a life saver. My lecture is super smart, but you seem to explain it in a more understandable way
This review well for me learning for others is great your sessions are well timed so never to long and easy to follow. I forgot how long these calculus problems can get. If I remember correctly some problems can get extremely long.
at 7:45 that MASSIVE 3!
U have a completely different face and life than I imagined when I listened to and watched your videos!
I have no word to thank you becoz I want give you very very very very big thanks.May God Bless You....
you didn't confuse us ...you just cleared all our confusions !
Omg... I actually got that.... if my University professor had attempted to teach me this then there is no way in hell i would have followed so clearly.
YOU ARE A MESSIAH..... Of my marks, thank you so much!
jazakum'ALLAH'UKHAYRIN! may, GOD bless you and I eveyone else watching these videos!
A good teacher makes a world of diference.
Thank you!
thank you so much for doing this!
i can never learn properly in school because my mind wanders a lot, and if i miss even 1 thing the teacher says, i won't understand the entire lesson!
but i can always replay videos and that's the best part
He does that, but in reverse order. Where the brown in the last line is the derivative of the large bracket, the pink times the green accounts for the derivative of the inner portion you mentioned: dy/dx (3x(-2)+(5x(3)-7x)(5). The light blue is the derivative of the last remaining inner portion: dy/dx (5x(3)-7x)(5). I was taught to tackle the outside first, so his method appeared a little backwards to me at first.
this is soooo much easier for me to understand...i havn't been getting good grades in calc and i think im finally getting caught up
Good at explaining with a lack of assumptions, and the mouse writing is clean too
Forever grateful to Sal!
I like the videos, but I find it much easier to think about it in terms of a function of a function (of a function). eg. f(g(x)) or f(g(h(x)))
For a simple one:
dy/dx = d/dx f(g(x)) = f'(g(x)) * g'(x)
For a more complex one:
dy/dx = d/dx f(g(h(x))) = f'(g(h(x))) * g'(h(x)) * h'(x)
For even more complex (like that example):
dy/dx = d/dx f(x + g(h(x))) = f'(x + g(h(x))) * (dx/dx + g'(h(x)) * h'(x))
Maybe not the best notation to learn first, but makes solving ones like that last problem easier.
Sal should make a playlist dedicated to teaching "teachers" to teach.
Howdy.
I'm 13 years old. I started taking Trigonometry a month ago, and now I'm here.
You must be thinking "Oh lord, you're humble aren't you?"
But hold on. This comment isn't about me, this is about Sal. I had the worst teacher for Algebra 1 in 7th grade, so I went to Khan Academy. I'm 13, and I'm doing calculus. Thanks Sal.
i actually followed through the whole problem, thank you.
You are explaining 20x better than my teacher . Keep it up good sir ! You're awesome!
this guy is dangerously good.
i think it would make things easier if we used substitution,for example after taking the derivative of the inner function we can write on the side that it is equal to some variable, do all necessary steps then reintroduce everything at the end.
Ok, my only question is, in the second example, when deifferentiating the largst bracket, why wouldn't you then multiply by the derivative of the inside? so where you have
3(3x(-2)+(5x(3)-7x)(5))(2), why dont i need to then multiply by the derivative of inside the bracket?, so, 3(3x(-2)+(5x(3)-7x(5))(2) X (3x(-2)+(5x(3)-7x)(5)
WHY DO I GO TO MY CALCULUS CLASS? I LEARN NOTHING. thank you soooo much!!! i'm passing calc because of you.
He's Indian. Such wonderful people. Very educated.
Not gonna lie, when you said that that example at 8:30 was going to be one of the harder ones we'd see on our test, I freaked out--you obviously haven't met my teacher. I hope you'll have examples with trigonometric and/or logarithmic expressions!
Thanks for all you do Khan Academy
Thanks for all your videos! Very helpful in exam period!
you're so much better than my current calculus professor, THANK YOU!
why did u distribute the -7 to the first expression instead of the second one?
ohhhhh
thanks
Actually quackhail6 is not correct sorry. You can not distribute the -7 across the second expression because everything inside the second set of brackets is to the power of -8. To think of it another way, the second half could be written as 1/(x^3 + 2x^2 - x^-2)^8: to multiply it correctly, the -7 would multiply the 1 making it -7/(x^3 + 2x^2 - x^-2)^8. Hope that is clear, its difficult writing it out without algebraic symbols and proper "1 over"
drpbrabbit False. 7 is outside the brackets and so can be used to multiply either term in final expression.
discuss21
Sorry, drpbrabbit is right... remember PEMDAS. You cannot multiply -7 into the second term until you take care of the -8 exponent. The only way to bring the -7 into the parentheses is to raise IT to the -8th power first THEN multiply it. That would hardly be considered 'simplifying' so it is factored in to the first term instead. If you still don't see it, try it with an easier example: 2(x)^2 is the same as 2 (x^2) but is NOT the same as (2x)^2... it is actually (4x)^2. You have to square the 2 first, then you can multiply it into the parentheses.
Khan is brilliant!
god bless you!
you make one who hates math turn into a math lover:)
congrats for finding a faster way in erasing:)
take care,WE need you.
4:20 Yay, not only have I learned something, Sal learned with me as well ^ - ^ and he's so happy about it...
This helps me more than my professor. Thankyou sooooo much.
Nice tutorial!
It really helped me with my homework :D
Thanks for sharing your knowledge xD
So.. so.. so.....good one of the best e-tutors....!!!!!😀😀😊😊😊🌀
WOOHOO! Thumbs Up! if your the 100,000 viewer! I love Khan Academy and these videos!
The last problem is incomplete, can it be simplified?
Jake4rmStateFarm
yus
Jake4rmStateFarm of curse but i don’t think anyone would want that except you are my calculus professor Wing. Lol
The example at 5:56 reminds me of INCEPTION, but with derivatives, anyway thanks so much as usual!