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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[name]

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.

[Arrogant_Artemis]

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

[Alex-nshj]

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!

[ricapp]

I had to study how to do derivatives, but it always mostly just memorization and I never understood what was going on under the hood of all those power rules, product rules etc.

This is the FIRST time I ever understood where it all came from, like what's the actual math under the hood! Thank you so much for your videos on this topic!

[notavailable]

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.

[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.

[SolidPayne]

I was so confused until the end. A derivative is basically just a substitution in order to find a specific point like a max or min.

[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.

[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

[kylecollins]

I learned more by the discussion about the tangent line than I have in four weeks of daily two hour classes.

[handsomememer]

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

[corsaircaruso]

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!

[adnkir]

You explained in eleven minute video what other tutors struggle to explain in one hour videos and still leave students confused. Thanks.

[roribabaaneneka]

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

[rimuruthegreat]

Definitely the best math tutoring channel on

[MAmanalac]

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

[jessicabaileyatsml]

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

[arialeen]

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.

[p.c]

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

[codingwithbm]

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