What is a derivative?

What is a derivative? Learn what a derivative is, how to find the derivative using the difference quotient, and how to use the derivative to find the equation of the tangent line. This is a free math video tutorial by Mario's Math Tutoring.
0:11 What is a Derivative
1:22 Finding the Slope Between 2 Points on a Curve
2:05 Difference Between the Average Rate of Change and the Instantaneous Rate of Change
2:34 Using Limits to Find the Instantaneous Rate of Change
3:15 What is the Difference Quotient
3:23 Notation for the Derivative
3:46 Example 1 Finding the Derivative of f(x)=x^2 Using Difference Quotient
5:02 Using the Derivative to Find the Slope at a Point
5:26 Writing the Equation of the Tangent Line at a Point
6:00 Example 2 f(x)=x^3 - 4x Finding the Derivative to Find the Relative Maximum and Minimums
7:40 Using the Difference Quotient to find the Derivative
7:57 Using the Binomial Expansion Theorem to Simplify
8:54 Setting the Derivative to Zero to Find Turning Points
9:35 Graphing the Polynomial With the Turning Points
10:07 Summary of What the Deriviative is, How to Find it, and How to Use It
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[@MariosMathTutoring]

If you would like to see another video I did on finding the derivative using the difference quotient click here: kzbin.info/www/bejne/rmnNXmeNjrGAbLs

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This guy reminds me of someone else

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@@chessplayer5728 nothing

[@sneakypress]

Good on you Mario. I have been studying lots of videos about differentiation, and finally with your tutoring, I understand what I was having trouble with. Thank-you. ☺️

[@name1799]

Within the first minute, you taught me more than my college level calculus class has ever. I still dont know wtf I am doing. I just know to follow the steps and somehow I get to the correct answer. I had no clue wtf a derivative even was, only told to find it. Thank you for explaining something that a college level course just expects of me.

[@jakejasperfox8368]

facts school likes to over complicate things till the point your brain is overwelmed and just blocks out all new information as useless

[@Arrogant_Artemis]

this is the best video by far i have found on derivatives thank you sir !

[@MariosMathTutoring]

You’re welcome! Glad my video helped you understand derivatives better!

[@kimcomorro2736]

I agree

[@MariosMathTutoring]

Thanks Kim! Glad you liked the video!

[@rickywilliams7397]

I came here just for a refresher. Ended up having a better understanding than before.

[@MariosMathTutoring]

Glad my video helped you Ricky!

[@tomzhangg]

Thanks, Professor X, godspeed.

[@notavailable453]

You did such a good job explaining this. I've seen so many cal students understand how to do the math but have no understanding of the actual concepts behind it.

[@MariosMathTutoring]

Glad you liked my explanation of what a derivative is.

[@chessplayer5728]

This guy reminds me of someone else

[@lesgolesgo5312]

@@chessplayer5728 EVERYWHEREEEE ahahahhhaa

[@Alex-ns6hj]

This is where I learned what a derivative was. Nobody ever explained it to me in one definition well enough. “A derivative is the formula for the slope of the tangent line” “The average rate of change is the secant line between x2 and x1”, which is where limits can come in. This is crazy helpful!

[@mashuli]

You sir, listen, I didn't even have to watch the whole video, you nailed it very quickly in less than 3 minutes. I was looking for a video I can use to help someone understand derivative, this video ended up helping me much better instead, job well done Sir, in less than 3 minutes. There a bunch of videos here just hitting around the bush and ending up with no real definition of derivative. Thank you

[@MariosMathTutoring]

You’re welcome mashuli! Glad you liked my straight to the point video about what derivatives are!

[@chessplayer5728]

This guy reminds me of someone else

[@lesgolesgo5312]

@@chessplayer5728 ok y every comment

[@chessplayer5728]

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[@lesgolesgo5312]

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[@yaakcon]

My son does not like to do the dirty work of researching and learning things so for time to time I help him by watching your videos. I must tell you that you are one hell of a teacher. You explain things very easily. Thank you so much.

[@corsaircaruso471]

Hey, there. This is a Doctor of Musical Arts who hasn’t taken more than 12 credits of STEM courses in his entire secondary education. I haven’t taken a single math course since an algebra review class in community college in ‘05, and no science courses since an Acoustics for Non-Majors course offered for Music students in ‘09. Of course I used some very basic knowledge of science somewhat in my grad level courses, but my main point is that I’m not especially educated in this area. I’ve never taken Calculus, nor Pre-Cal, nor even Trig. I barely remember high school geometry. This video helped me get a significantly better understanding of the basic mathematical use of derivatives in an abstract sense. Some of the algebra was a bit fast for me, but I understood somewhere between 85 - 90% of what was discussed here. I’m pursuing this because I’ve always had a conceptual interest in physics, especially quantum physics and general relativity. PBS Spacetime and similar channels helped feed my interest, but I felt I lacked any understanding of even the basics of the equations. Channels like this are helping me continue my education in mathematics so I can better enjoy Newtonian, Lagrangian, and Hamiltonian mechanics, so I can better comprehend the reasons why certain quantum phenomena are expressed they way they are by the math, or how the energy needed to accelerate mass to c approaches infinity. Thanks so much for this excellent resource!

[@nonsenstronaut5477]

I've been studying this topic for years now but this is the first time that I have genuinely understood the nature of a derivative. Thank u Professor X!

[@handsomememer]

Can we just take a minute to respect this guy's amazing ability to legibly write/draw on a whiteboard

[@dittery]

I always learn concepts like these but my professors never actually explain them as a concept, they only talk about the formula and how to use it to solve problems. This makes math really boring because you have no idea what you're actually doing. Learning HOW and WHY stuff works in math makes it 100x more fun.

[@roribabaaneneka]

This is probably the best derivative explanation I have watched. Thank you so much sir.

[@MariosMathTutoring]

You’re welcome Wilhelm! Glad you liked my video!

[@minimhlk7254]

Never thought the sword demon will study the derivatives XDD

[@TheDankDev]

Thanks, I find that understanding the concept behind something makes doing math problems so much more logical. I'm loving my AP calculus class so far!

[@p.c2750]

You are one of the very mature mathematicians i know. Thank you for your conscientiousness!!!

[@spidertyler]

Took me till halfway thru the video to understand since I haven't taken pre-calc yet. But this is the only explanation that I could understand on KZbin. Thanks for this wonderful video!

[@jessicabaileyatsml]

I went from being completely lost in calculus to understanding the how and why with just this one little video. Thank you sooooo much!

[@deltaexplorer47]

AWESOME explanation Mr. Mario - Thank you very much !

[@jeremiahhyatt6935]

Thank you so much! Concepts like this beat being shown formulas with no explanation any day.

[@codingwithbm]

Thank you! You explained it very well. I understood everything, even as an 8th grader.

[@MariosMathTutoring]

Glad it helped!

[@welcometotheshow5247]

What a beautiful explanation, I really like what u did with the graph thing, very insightful and very resourceful.

[@paulaan881]

Best video hands down! You saved me.... mostly in this pandemic. Thanks teacher

[@user-sp1of1ic3h]

Thank you Mario, for the concise and thorough review!

[@arialeen5329]

Online class is a struggle for me who has experience this kind of set up for the first time. Thank you so much for this video, sir! It's so clear and precise.

[@chessplayer5728]

This guy reminds me of someone else

[@basilhawamdeh593]

it's the second video I watched for you, I feel like you've done some magic for me to fully understand the idea, you're really great sir! thanks a lot.

[@MariosMathTutoring]

You’re welcome Basil! Glad my videos have been able to help you!

[@radoslawcecherz1161]

i share your calm with the topic - fantastic style of sharing the knowledge

[@Ali_azimi]

I would say that HE has been the best teacher for me so far!! Appreciate you man!

[@jamesstark4136]

Wow okay thank you. I came to this video because while I could find the derivitive I had no idea of what I was actually finding and needed this information to understand how exactly the math works. Thanks A Ton!

[@MariosMathTutoring]

You’re welcome James!

[@luanajoseph5537]

This video helped me understand derivatives better! Thank you Sir.

[@DC-qs8nc]

Amazing job explaining everything!! Thank you!!

[@memesthemaker8429]

Thank you so much! I had difficulty understanding derivatives but this video came to the rescue

[@idreesalao6155]

Really helpful explanation. Thanks for this, much appreciated.

[@wahtdafukkkkk1182]

I just spent the last semester in college algebra wondering what the point of all the f(x+h) and fog was. I also wondered for the entire year how exactly you find relative max and minimas of turning points. This video just connected so many dots for me, thank you!

[@MariosMathTutoring]

You’re welcome! Glad it helped!

[@shokhmurodkodirjonov1457]

Your English is really perfect Sir! I got what derivative is. Thank you Sir

[@MariosMathTutoring]

You’re welcome! Glad my derivative video helped you!

[@ashleycampbell613]

Perfect - thank you. This is much better than written text books for sure

[@rimuruthegreat4027]

Definitely the best math tutoring channel on Earth!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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Glad to see Jhonny Sins back as a teacher. Can't wait for your next class! 🥰️

[@santiagocos4769]

Ha. I had the same thought.

[@sweetdaddyg950]

I had no idea why I was finding derivatives or how they were used, and this made EVERYTHING make sense. Thank you so much for this video explanation!

[@MariosMathTutoring]

I'm so glad!

[@parwaiznoori]

Thank you for the explanation sir! It helped me a lot.

[@nani2155]

Thank you very much, I have an exam tomorrow and this helped me very much. I have finally grabbed the concept of derivative

[@timvogele3884]

Great video. Love the explanation of why this is useful.

[@zadramm1]

Sir you are Amazing ! you know why you are amazing? because you make thing so simple.. why something is so ... and you explain it so simply.. thanks so much.

[@MariosMathTutoring]

Awesome! Glad you are liking my videos and my channel Zadran!

[@Indianfootballteamedits11]

What a great teacher....👏👏

[@saalarfaisal1792]

Excellent Video. Great Explanation!

[@alexwilliams7063]

This is truly a mind blowing video, I was struggling with the concept of derivatives but now it all makes sense! Thank you!

[@MariosMathTutoring]

You’re welcome! Glad my video helped you!

[@cal0729]

By far the best video I’ve seen about derivatives. Thank you

[@MariosMathTutoring]

Glad you liked the video Callum!

[@annzhang6958]

Great teacher, thank you!

[@vanessachen2330]

Thank you very much for posting the helpful videos, and your videos are alway in my consideration of wtching whenever I am stuggling for math problems.

[@nathanclarke7162]

This video is really good, now I understand what a derivative is and does

[@Venyrean]

miraculous explanation, can't be more satisfied !

[@animishsukhwal]

Superb explanation !! thanks

[@jacobsamuel8851]

Thank you sir,it was wonderful explanation

[@johnmutiso8357]

great explanation

[@normanhenderson7300]

I watched this video so many times, such that each time I watch it, I discover another aspect not previously seen.

[@MariosMathTutoring]

Glad you liked the video Norman!

[@pbrooks4040]

Great vid. Got a real-world application of the use of a derivative? It can be computed, but so what? What do I do with it?

[@MA6manalac]

After rereading and rereading the textbook and watching multiple videos ive found the one that actually teaches what I need

[@MariosMathTutoring]

Glad my video helped you Marc!

[@normanhenderson7300]

Even after purchasing a textbook of my own- Calculus for Dummies, my understanding is less than expected.

[@jcbgoncalves]

Excelente explicação!

[@ogabekhamdamov3179]

Thant you for all your videos Mr Marios .It helped me a lot to understand what Derivatives is !!!

[@MariosMathTutoring]

You’re welcome! Glad my videos have helped you!

[@abodaher6808]

Thanks Amazing explanation

[@ramonleal240]

Currently writing a bullsh*t essay about derivatives and this video helped a lot, thanks Mario!

[@ishaanarya7408]

Best video for Deriviatives available on KZbin. Just a request as well as suggestion Professor: Please write a little larger in size, it will make contents visible efficiently. Otherwise you just nailed it. Loved it.

[@MariosMathTutoring]

Thanks Ishaan!

[@hackingnut4164]

If I were really good with math I would have created a time machine... Then I would have went back in time and got my younger self to meet you and learn math from you and then I would be better at math. Then we would have met as two professors in an alternative time line of our lives and we would have been great friends teaching math to people who don't have a time machine. I also have ADHD so this was what I thought of after watching your video. Thanks!

[@marnierogers3931]

10/10 video, thanks!

[@foxhound-0]

great respect . The way you teach but please one thing put the definitions for us and the different symbols of derivitives ill be waiting for you please and thanks a lot .

[@ssn8]

Thank you so much sir

[@AnNohor]

Sir, you are a magician, lots of love, respect and dawah from Bangladesh.

[@MariosMathTutoring]

So nice of you!

[@yarrowstudy3530]

Wow!!! Thank you sir

[@rameshwarpawar1830]

Very nicely explained sir

[@amourito123]

Thank you for the explanation! It was clear and coherent

[@MariosMathTutoring]

You’re welcome!

[@jaretlesure6151]

Thank you! It finally makes sense!

[@MariosMathTutoring]

You’re welcome Jaret!

[@ShinsDomain]

Loved it a lot 💕💕

[@hahahaha9224]

Thank you sir!!. I knew you are a real teacher aside from being a doctor, plumber, a soldier and an astronaut.

[@shanegepilano6442]

Really helps me a lot! Thank you!

[@MariosMathTutoring]

You’re welcome Shane! Glad my video helped you!

[@halland7892]

Best english video about derivatives

[@dijonstreak]

AWEsome dude. boy. thanks to YOU. i FINALLY got it in Plain English very well explained. thank YOU SO much. !!!!

[@MariosMathTutoring]

Glad it helped!

[@ali_aldur]

Thank you very much sir, your follower from Iraq. Very useful video thanks much.

[@MariosMathTutoring]

You’re welcome Ali!

[@gaspinghorse4865]

Thank you very much for the demonstration, I was struggling to figure out what a derivative actually meant before watching this video.

[@MariosMathTutoring]

Glad it helped!

[@TT-gp2os]

This video helped me understand why we need to use derivative in ML, Thanks great job

[@MariosMathTutoring]

Glad it helped!

[@grins047]

Thank you very much. These videos help.

[@MariosMathTutoring]

Glad to hear that!

[@geraldillo]

excellent video! 🏆

[@phyomyint2750]

Thank you so much for this video.

[@MariosMathTutoring]

You’re welcome!

[@huilovtsam_kaput]

The increment of the function can also be expressed through the difference of functions, using the factor k-->1: d(f(x))= f(kx)-f(x). Similarly, the increment of the argument: dx=kx-x. Then d(f(x))/dx=lim[k-->1] (f(kx)-f(x))/(kx-x)

[@arunkhare7331]

nicely explained

[@fruitmix4health]

thanks for this great explanation

[@MariosMathTutoring]

Your welcome! Glad it helped!

[@rosajijo7713]

Thank you sir it was a great help

[@MariosMathTutoring]

You’re welcome Rosa! Glad my video helped you!

[@walterbaldemor2128]

Derivative of x^2 is 2x , same as nx^(n-1), now i know where it came from. Thanks sir mario.

[@MariosMathTutoring]

You’re welcome Walter!

[@yassineaghlaoui1906]

They used to teach us in school this way: whenever we tell you the derivative of 5x^2 you answer 10x. Without knowing where it came from, we'd never advance and understand real maths.

[@MariosMathTutoring]

Glad my video helped you understand the origin of those shortcuts for finding the derivative!

[@yassineaghlaoui1906]

@@MariosMathTutoring Can you please make a new category of videos about how to revise maths and how to be more familiar with it? It'll be an awesome gesture from such a great teacher.

[@ulugbekkirgizov452]

amazing explanation.thank you

[@MariosMathTutoring]

Glad it was helpful!

[@northernlight1000]

Thanks!!

[@Ahmed-vs1ui]

You are the absolute bestt

[@phattran4858]

your voice is really clear, thank you

[@MariosMathTutoring]

You’re welcome! Glad you liked the video!

[@zhouhang5330]

That is too clear to not understand derivatives. Thank You!

[@josephls4038]

I love it

[@markwojciechowski6435]

thank you so much

[@MinnieNinnie]

This is amazing! Thank you:)

[@MariosMathTutoring]

Glad you liked the video Nikolina!

[@hiwot7668]

Very very very helpful

[@andrekasamuel3987]

Amazing explanation!

[@MariosMathTutoring]

Glad it was helpful!

[@AlexSmith-dh1oz]

Thanks Mr. Clean

[@azozvn7014]

thanks a lot man

[@handokosupeno5425]

Thanks mario

[@avineshbenjamin1782]

Where were in my +1, +2 Sir Superb video...😃