The Derivative - The Most Important Concept in Calculus

The derivative is one of the most fundamental and powerful concepts in all of mathematics. It is the core idea behind calculus and serves as the foundation for countless applications in physics, engineering, economics, and even biology. But what exactly is a derivative, and why is it so essential? In this video, we take a deep dive into the meaning of derivatives and why they are at the heart of understanding change in the world around us.

At its core, the derivative measures how something changes. It tells us the rate at which one quantity varies with respect to another. Whether we are looking at the speed of a moving car, the way a population grows over time, or the slope of a curve at a particular point, derivatives provide a mathematical way to describe and predict these changes. We will explore several key examples, including how derivatives help us understand velocity and acceleration, how they reveal the steepness of a function at any given point, and how they allow us to optimize real-world situations like maximizing profits or minimizing energy usage.

Beyond its practical applications, the derivative represents a fundamental shift in mathematical thinking. It allows us to move from static quantities to dynamic systems, where we can analyze how things evolve rather than just where they are. By the end of this video, you will have a deeper appreciation for why the derivative is not just an abstract mathematical tool but a critical concept that shapes our understanding of the universe.

[ChrisB]

Very old guy here - despite having an engineering degree - never (being honest!) got to grips with calculus such I felt any degree of decent understanding - this really has helped a lot. Never to late to learn. Appreciate this video and look fwd to next - maybe I'll die happy :) :)

[nathanjohnnie]

There is none like Professor Gibson, when it comes to the variety of topics that he covers. Absolute beauty.

[lovefly]

Calculus got me out of a speeding ticket. In court, the cop said I was “driving at a high rate of speed.” I told the judge I was driving at a constant speed using cruise control, hence my rate of speed was 0. He asked to elaborate, so I went on Calculus mode. The judge just gave me a smile and canceled the ticket, he said he’s never heard that one before.

[amirlaplap]

I always add your videos to my education playlist when they are added. As a PhD student in electronics, most of these concepts are familiar to me to some extent but the method of teaching and explaining you implement stems from deep thoughts and provocative ideas. They are not just copying the content of textbooks on the whiteboard. You have critically studied them and thought deeply about them from as many aspects as possible and after truly assimilating them you share them with us with your own intuitive method. Kind regards, Amir.

[bernarddoherty]

Great great teacher! Be sure to tell the students that it’s called a ‘derivative’ of a function simply because it’s ’derived’ from the function.

[tonykimani]

My favorite guy already overtaking Organic chemistry in his way of teaching 🎉

[shujazaidi]

You are doing a great service Professor. Your passion for transfer of knowledge to others just pours through in every lecture. Thank you and keep it up!

[Badddi]

The fact that is one hour video was great, love these complete explanations. Your rhythm is great and the time goes by easily.

[siddharthpatil]

i can start understanding calculus now (plz continue this series) 👌👌

[Farjah-qp]

Where were you when I was struggling to understand the concepts of derivative and integral 57 years ago. Looks so easy now the way you teach it; later in life I had to teach myself calculus in order to build and explain physiological models.

[jppm]

Next semester i go to college but I'm in my late 20s this man is very good at explaining everything I'm not good at math but i take engineering what a suicide but i understand what math is in his videos he is good at explaining to beginners you can get it even how hard it is the way he explained everything more videos to come sir ❤ I am always watching.

[Mekinhumbel]

The speed at which this stuff is starting to make sense to me is double prime cubed :). Keep 'em coming! I see why there are 1.5 M subscribers

[السرطاوي-دب]

Really appreciative of this breaking down to how we humans can think creatively.

[suribabubonu]

Thanks for another amazing concept. Please do more videos on calculus and rocket science 🎉

[Badddi]

Man, that was awesome, the more you explain the more i want to learn. Big thank you from Brazil.

[lolwuttup]

Wow you’re still putting out educational videos? Good on you man!

[kitsurubami]

Thank you!! I already know derivatives well, but I love calculus and I love your lectures. Appreciate you!

[kersi-sandiego]

Jason, another gem, even though I've watched it many times before.

[AlbertoApuCRC]

NO puedo creer que solo tengas 35 likes, Muy bueno el video

[kabandajamir]

The world's best teacher thanks sir