Derivatives... What? (NancyPi)

MIT grad shows the DEFINITION of the derivative and how to FIND the derivative using that limit definition. To skip ahead: 1) For what the derivative MEANS, skip to 0:23. 2) For the SLOPE OF THE SECANT line, skip to 2:53. 3) For the SECANT BECOMING THE TANGENT and DEFINITION OF THE DERIVATIVE, skip to 5:54. 4) For how to FIND THE DERIVATIVE USING THE (LIMIT) DEFINITION of the derivative, skip to 10:28. Nancy formerly of MathBFF explains the steps.

INTRODUCTION to derivatives:

1) WHAT IS THE DERIVATIVE? It's a function that tells you the slope (of the line tangent to the curve) at every point. Another way to think of it is that the derivative gives you the rate of change at any instant (the "instantaneous rate of change" at each point). You can find the derivative either with the proper DEFINITION OF THE DERIVATIVE ("by the limit process") or the faster, simpler way with the shortcut derivative rules such as the Power Rule, Product Rule, Quotient Rule, and Chain Rule. This video shows the first way, with the DEFINITION of the derivative, and how to use it to calculate the derivative.

2) SLOPE OF SECANT: To make the definition of the derivative, we can start with the slope of the straight secant line through a point x and some other point nearby, h distance away horizontally. If we label the two points and use the slope formula to write an expression for the slope of this secant line, the expression you get is [f(x+h) - f(x)] / h, which is also known as the DIFFERENCE QUOTIENT.

3) DEFINITION OF DERIVATIVE: That straight line slope we just found is actually a decent ESTIMATE for the slope at x, but it's not really the slope at x. We can get the EXACT slope of the tangent line at x by closing in on x and narrowing h to zero, by taking the LIMIT of the secant slope, as h approaches 0. The slope of the secant line becomes the slope of the tangent line. Not only is this limit equal to the slope of the tangent line, it is the definition of the derivative (when the limit exists) or f'(x).

[jaguarwolf]

For the first time in my life I know what derivative actually means.... Great Job👍👏

[wishyjoshy]

It takes true mastery of a subject to make it as simple as you do. Super grateful! Thank you!

[RR-mphw]

It's great to see someone thinking through what they are explaining. The sign of a natural teacher.

[sibbyeskie]

Nancy is by far the best at explaining these fundamental concepts. Perfect amount of repetition, visuals, analogies, and pacing. And clean. I always struggle and then remember to check if Nancy explained it and *boom*, I understand it far deeper.

[sarahmt]

Girl, you make everything so much easier!!!! Thanks - alot

[HL-dwdl]

I am from India. Thanks a lot for teaching calculus in a simple manner. All Indian teachers on youtube limit their content to stuff that is asked in exams rather than teaching for knowledge.

[sebastianpozziazzaro]

Finally! The clear, half graphical, half algebraic explaination I was looking for. Thanks for the clarity and the consistency!

[sonasunny]

I wonder why she isn't making videos often. Precious things are always rare. Please make more videos nancy you really are a mathsaver.

[waynethurman]

Had so many college teachers who mistakenly thought their purpose was to weed out instead inspiring to learn and giving students confidence toward that goal. Einstein said that the poor teacher talks the simple and makes it difficult, but that the good teacher takes the difficult and makes it simple. Nancy is the best at making math understandable that I have ever seen; ever!

[nicolerodrigues]

you're literally saving my life. SO clear. My textbook SUCKS and is SO confusing. Thanks for helping me get through my homework. You explain things clearly and are not boring. Appreciate it.

[Alex-hpnl]

my gf wants me to watch a different math youtube channel bc this one makes her jealous.

[revazzhvania]

Very few people explain concepts in an understandable way. I mean, when you ACTUALLY UNDERSTAND what it is and why it is there. It's easy to just DO IT because you saw 100 other examples but, what it really is, is what matter the most. This was amazing.

[davida]

This video was recommended to me.... and although I'm no longer in my calculus II course, I become a bit nostalgic of the times I would stay up late watching your videos to solve just a few questions... ah good times! Thanks for your help during many countless efforts at my homework and exam preps!

[erekadivine]

God bless this lady. I’m so glad I tumbled upon this page my calculus teacher at the university explains things in the most complicated way possible, even if you understand you’ll end up confused. But THIS HERE IS GOLD!!thank you you soooo much❤

[michaelmorrison]

Nancy, your math videos are good refreshers for our senior engineers, especially when dealing with complex transient problems. And several of our younger production employees are interested in engineering, and maybe pursuing engineering and your tutorials are an excellent means to motivate them. Thank you.

[vectr]

Hey there, I completed my engineering an year ago... and just on this random day I remembered how this channel helped me out in my first year. Thanks a lot!
Lots of love, stay safe and healthy. I'll never forget you. Thank you. ❤

[georgevladimirovich]

Thank you Nancy. I am taking your videos as a reference each time when doing math with my daughters and they have both became good at it. The school marks are also confirming that. I appreciate your work and your commitment. Thank you.

[srinip]

That's great! At 65, I had completely forgotten the first principles derivation, and thank you for reminding me!

[cmdrbobert]

That was incredibly helpful, I just had my first lectures on derivatives a couple days ago, and this part was breezed over as if it were assumed knowledge. Thank you for breaking it down.

[robnee]

Your lectures simply show me how well you understand what you are talking about! It's a real gift to be able to be so clear. Thank you.