Understanding Differentiation Part 1: The Slope of a Tangent Line

The first operation in calculus that we have to understand is differentiation. So what is it, exactly? Well there are a couple of ways of looking at it. The first one involves finding the equation of a tangent line for some point on a curve. We can't do this with algebra, because for that we would need two points on the line, and we only have one. Enter calculus!
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[@RajeshSharma-tf2xx]

Excellent explanation. If you can believe, I have been looking for this kind of explanation for 3 decades

[@CreativeHandleOrSomething]

This is so incredibly helpful for me! I'm on my first semester of Chemical Engineering, and my math is, to say the very least, lacking. Professor Dave's classes on top of my Teacher's is just that extremely needed extra push. Reviewing stuff, taking formulas... I always come back to this channel for both! And it's not just calculus! Chemistry (even the lab stuff), physics... There's everything you need and with awesome explanations! Personally, even the way Prof. Dave talks (in a non rushed, clear way) is very helpful, considering English is not my first language. Thank you so much for what you do, professor! You're honestly so good at what you do!!

[@ibrahimjaiteh42]

I also want to study chemical engineering. How is university?

[@kavyahegade6477]

Who are you? You don't seem to be an ordinary person.. you're amazing and extraordinary 🙏🤝

[@destinyovbiebo8988]

He is a guru

[@rosepierce9382]

he's Jesus.

[@nikolaigrut]

I like your way of explaining it. I had calculus some 56 years ago and was mediocre. It’s nice to revisit.

[@harisaisenthilkumar2880]

Wow you are a very old person! Wonderful to know of such people

[@Daltnation]

@@harisaisenthilkumar2880 chill

[@rosepierce9382]

@@Daltnation no, you chill

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@@harisaisenthilkumar2880 true in India it's too uncommon 😲

[@harisaisenthilkumar2880]

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[@ezazq9876]

Man I m an engineer, but after watching your videos, I believe that I had a myth that I am engineer. Tx a million for this initiative, it's nothing short of a boon

[@justlearning529]

Thanks so much for taking the time to explain through a comprehensive course. Great presentation and visuals have really helped me understand!

[@nehalsingh3094]

you are one of the best teacher I have studied from because your videos are not complicated and you explain in balanced pace😄 Thanks for all of your videos 😃

[@mranonymous_25]

Next: how to play professor dave theme on piano

[@mranonymous_25]

@Mathias Fernandes lol😂

[@somerandomguy4075]

@@mranonymous_25 s

[@radiotv624]

I love these videos! Never stop!

[@jeffreyquinn3820]

I had a grade 12 teacher who taught this very well, but 34 years later, I'm doing another degree in the sciences part-time. I probably won't need calculus for a little while, but it's never too soon to review. I think the problem with some first and second year math and science classes is that they try to fit too much material in too fast, so students don't absorb all the material at the beginning.

[@ivan_ivan0v_]

Great video man! Thank you for explaining all these things to us!

[@ronin6158]

one of the most succinct and clearest explanations

[@heli35]

You are marvelous Prof Dave!

[@MIZORAM_mafaka_hnamte]

_Today's students are so lucky_

[@MrBroncolinShows]

MIZORAM - Mafaka Hnamte Internet is heaven!

[@annyan904]

indeed i totally agree with you

[@neurodivergentsophie8411]

Yep, professor dave is here to save me at all times!

[@sidtronics]

But these students take it for granted

[@Rani-mt7dj]

But we have lot to learn than you did so just stay silent

[@rp2804]

This video is better than anything out there

[@jennakuuki]

thanks for your videos! i'm gonna tell my friends about this channel

[@gaminisiriwardana5946]

excellent way of explaining. I got amazed with your explanation and did lot of calculus sums. Sir I will be sending very special 4 questions through email to you. Please be kind enough to reply. Interestingly waiting for your explanation . Thanks

[@alejandroalmarza8447]

Excelent explanation profesor very clear thanks

[@muhammada.muhammad6287]

extremely good job

[@minhazulabedinakib9429]

You are great sir!

[@aksharasbhat6480]

Superb sir I love listening to you

[@anthonypeak1]

thanks dave!

[@nipunsharma7418]

So to study calculus , we should know algebra and trigonometry and polynomial

[@sibalogh]

It seems like it, eh? :)

[@pinklady7184]

What is polynomial?

[@destinyovbiebo8988]

Algebra is necessary for cal

[@marcusvergara6193]

@@pinklady7184 if you don’t know what a polynomial is, then don’t be here.

[@pinklady7184]

Skillertrap101 X lolol, my ancient comment above is a year old. *VERY OLD* comment. I have since been self-studying mathematics and I have pulled up much. I am now reading a book "Calculus, Metric Version" by James Stewart. After that, I will level up to calculus II and III. I am also reading books on real analysis, and linear algebra. I will shortly move onto topology and abstract algebra.

[@RanjeetKumar-le7vz]

thanks so much sir

[@stan1677]

Quality education!

[@santoshgujar5237]

✨Thank you, Sir, 🙏🌺✨you are such a beautiful Teacher🙏🏼🌺✨

[@AmanSingh-db3nr]

Sir please make video series on complete topics of differential equations 😊😊♥️♥️

[@mareenabowman7495]

Bless you

[@barnabasabatu2407]

Thanks!

[@saikrishna8871]

Iam waiting for this many days

[@rajns8643]

Which character is ur avatar man? Onedera?

[@sarikachaturvedi4284]

Hate to be that person but it is I * was or I * have been,

[@jutiarhasan]

Thanks for this nice and useful explanation, What program is been used to make this video? Thank you for answering me

[@ProfessorDaveExplains]

adobe after effects

[@jutiarhasan]

@@ProfessorDaveExplains thanks

[@topiado2073]

Sir nice.

[@Shenshenzpoo13]

professor you deserve more subscribers!!

[@ProfessorDaveExplains]

i agree! tell your friends :)

[@zainabfatima9736]

what a reply !! 😂😂😂😂👍🏻 well played !

[@jorgeignacioillacoyo4774]

@@ProfessorDaveExplains Hola

[@destinyovbiebo8988]

He really does

[@katelynmiller2947]

could you make a video series on how to do differential equations?

[@ProfessorDaveExplains]

that will be coming soon!

[@rassimsimou1594]

Good

[@lidyaangelin8927]

How we initially got the basic line equation.... Parabola equation etc.... Can u make videos on how various equations have been formed .. Example : how y=mx+c for straight line...

[@ProfessorDaveExplains]

i have over 100 math tutorials before this one in the mathematics playlist that explain everything about lines, conic sections, etc. check them out!

[@lidyaangelin8927]

Professor Dave Explains thank you sir... I will def go through it...

[@pawanyadav3399]

Nice

[@mffmart]

For the record: I understand he was explaining the relationship of limits of infinity and calculus, but this is how I would go about this problem in real life. I find full simplification easier to understand, so this is what I noticed: expand the binomial on the numerator by turning x²-1 into (x+1)(x-1) and then divide the numerator and the denominator by x-1; this leaves you with m=x+1. Adding one to the x-value and getting m=2 could have been a simpler way to approach this, although I know this specific problem was not the scope of the video. Just here in case anyone wants to take advantage of the info. 😊

[@AustinMulka]

This is episode 1, slow down buddy.

[@mffmart]

@@AustinMulka 😂😂

[@rosepierce9382]

Hey, Miguel! Can you tell me why you're adding 1 to the x-value?

[@GhostSamaritan]

@@rosepierce9382 Because if x is 1 then x-1 is zero. Then we can state that, as x-1 approaches zero, the m value approaches 2.

[@shahebasweety2235]

Hey professor make videos about human physiology ur amazing

[@ProfessorDaveExplains]

anatomy and physiology is coming very soon!

[@telepcanin2878]

so the derivative of x2 is 2 or 2x?

[@leadarchitect7799]

2:19 but why we are moving the second point professor please explain

[@ProfessorDaveExplains]

to get closer to the first point

[@abhcy]

Alright so we're the pre requisites of Calculus are > Functions ▪Exponential Functions ▪Logarithmic Functions ▪Equations of Straight Lines > Coordinate Geometry ▪Slope Formula > Trigonometric Identities > Basic Geometry > Algebra ▪Polynomials ▪Quadratic Equations ▪Binomial Theorem > Number Theory ▪Number Systems ▪Complex Numbers Correct me if I missed anything

[@natu.d9546]

2:10 Couldn't we just take the first derivative of y = x^2? Then we'd get 2x, which we can take the value of m to be 2, then we plug in the points and get the same answer Y = 2x - 1. It kinda gets the job done pretty quick don't you think?

[@compresseddepression7403]

that doesn't really explain differentiation though does it....this way at least shows the concept of approximating when the change becomes infinitesimally small and how you can actually justify getting the gradient from only one point on a graph which logically doesn't really make much sense once you think about it (how can you measure "change" if you only measure one point)

[@annyan904]

really i agree with you today's students are indeed very lucky

[@destinyovbiebo8988]

Yeah, we didn’t get during our high school days

[@Entropicalli]

Great video however consider using an example other than one. It is hard for beginners to see exactly what is happening with one - one squared is one so it's obscure. Just a suggestion feel free to ignore.

[@jamiemedalla6326]

professor, how do you make those videos? do you have app to make those animation

[@ProfessorDaveExplains]

adobe after effects

[@tGoldenPhoenix]

Done.

[@fyori7376]

4:25, how did he figure out that b equals to -1?? Please someone, answer my question

[@fyori7376]

I think I just figured it out. I didn't notice that he plugged in the (1,1) into the equation

[@rustyspottedcat8885]

... in short ... if there is a function f: x -> y ... there MAY be a function ... f ': dx -> dy ... if so ... f is integral of f ' ... and ... f ' is derivative of f ... the logic behind is simple ... if f is bijective function of each POINT x to y ... then there MAY be bijective function f ' of each SEGMENT dx to dy ... so ... f is point to point function ... f ' is segment to segment function derived from function f ... in the infinity of arbitrary small segments it becomes differential value ... in Leibniz notation known as "d" ...

[@jorgeignacioillacoyo4774]

Disculpe profresor me encanta sus videos un favor sera que puede poner subtitulos en español porfavor

[@domenicodefelice2083]

Puedes habilitar los subtítulos automáticos en español. No son perfectos, pero deberían ser lo suficientemente buenos. Debajo del video, habilita los subtítulos haciendo clic en el ícono "CC". Luego haga clic en el ícono del engranaje, haga clic en "Subtítulos / CC", seleccione "Traducción automática" y luego seleccione "Español".

[@totalfreedom45]

​@@domenicodefelice2083 ¡Excelente respuesta, Domenico! 👋 Seguí tus claras instrucciones pero en mi caso la traducción del ruso al inglés de un video corto en KZbin es *_malísima._* 😂

[@AmanSingh-db3nr]

Sir I want to ask you one question- "what dx in integration notation mean"

[@compresseddepression7403]

pretty sure it just refers to an incredibly small change, dx being change in x,

[@jollyjokress3852]

Does a line perpendicular to the curve that goes through a certain point also only share just this one point with the curve? Dont know -probably not ;)

[@ProfessorDaveExplains]

so the word perpendicular can only apply to two lines, not to a curve!

[@jollyjokress3852]

Professor Dave Explains okay thanks! Beyond my scope of understanding momentarily but thanks!

[@ProfessorDaveExplains]

so perpendicular means two lines have a right angle between them, but a curve is changing direction constantly, so we can't define an angle between a curve and a line

[@jollyjokress3852]

Professor Dave Explains new day, new brain.i get it. If you'd represent a curve just by aligned dots one would notice that if a line that had the exact thickness of the dot would always cut one of the other dots because theyre kind of laterally in front of the middle dot. But what about a curve that changes direction? Haha! Thanks!

[@ProfessorDaveExplains]

whoa, sorry i don't understand at all! all curves change direction, if they didn't, they'd be lines.

[@caiomazza7489]

I dont get why y'=2x-1 would be the derivative of y=x^2, becaus if you apply the rules of diferentiation to y then the result woulde be 2x, can someone explain?

[@user-ws4ld6ek7q]

You're correct: f'x: x^2 ==> fx: 2x and we have know that (y-intercept) at x = 1, m = 2, Equal to 1 (as showing in video @ 4.13min) y-intercept = mx + b y-intercept = 1 m = 2 x = 1 b = ?? (unknown) So: 1 = (2)(1) + b doing basic algebra: b = -2 + 1 = -1

[@ChromeandClean]

@@user-ws4ld6ek7q why di you change 2 into -2 at the bottom

[@compresseddepression7403]

because y = 2x-1 isn't the derivative it's the equation of the tangent

[@aselim20.]

I watched it.

[@leadarchitect7799]

2:50 professor why you are taking point only less than 2 not above 2

[@ProfessorDaveExplains]

because the point is to get closer and closer to the first point

[@PaulisM-e7]

0:49

[@albertodelrivero2426]

Te rifaste krnal

[@nipunsharma7418]

And to be theoretical physicist we need to know calculus algebra and trigonometry and geometry

[@ProfessorDaveExplains]

probably more than calculus, but get calculus down first!

[@Gd_Monsterforce]

You were making a limit table(obviously).

[@science-y9209]

Woah

[@lestahass8849]

But the derivative of y=x*2. Is 2x not 2x-1 Please anyone

[@David-mm6nx]

Do you mean the derivative of x²? In that case, it would be 2x by the power rule: power rule states the derivative of x^n to be n(x^(n-1)) where n is the exponent of x: d(x^n)/dx = n(x^(n-1)) Therefore, d(x²)/dx = 2(x^(2 - 1)) = 2(x^1) x^1 is just x: d(x²)/dx = 2x The case of 2x - 1 is not the derivative or slope of the line. It is in the linear equation of: y = 2x - 1 This equation is simply the tangent line's slope-intercept form. The slope of this line will be the derivative of the point of intersection of y = 2x - 1 and y = x² ( (1, 1) in this case): d(x²)/dx with x as 1 = tangent line's slope = 2 this is proven with d(x²)/dx = 2x (the equation with the power rule): plug in x = 1 d(1²)/dx = 2(1) = 2 // same answer

[@user-kg2ks8vk9z]

Secant line is a straight line, it must have a constant rate of change or slope. Isn't it?

[@compresseddepression7403]

yea if it didn't it would be a curve

[@luctiber]

😀😀😀😀😀

[@shlokbajpai4232]

Hmmm... Professor Dave, I think you have made a mistake. Isn't the answer 2x, instead of 2x-1? d/dx(x^2)... so you bring the 2 down making it 2x, and then substract a power, so 2x^(2-1) = 2x^1 = 2x. Please let me know if it is I who has made a mistake, or if the mistake is yours. Sorry if I sound harsh, I definitely don't mean to :)

[@compresseddepression7403]

The answer is 2x-1 because that's the equation of the line which was the thing we needed to find, if you just differentiate the curve you will just get the gradient function for the curve which will be 2x.

[@ethanlee2602]

Jesus teaches Calculus. I love it.

[@sabriswish]

ok boss but why do u want to find the slope of a tangent line?

[@ProfessorDaveExplains]

to find the rate at which something is changing.

[@sabriswish]

thanks boss

[@sabriswish]

my professor skipped this part

[@user-kx7lz2qz7e]

KFUPM in Saudi

[@shagan47]

Our good old friend y=x^2😅

[@lits0_042]

This isn't as complex as I thought

[@jesusmeneses2771]

It's hilarious because i don't speak English so well and I learn better than spanish videos