Definition of the Derivative - Calculus

Derivatives part 1. Definition of the derivative lecture from our introductory calculus pilot course. First of many animated videos to come from http://www.free-academy.com. Why read your textbook when you can access our calculus videos? Help support tuition free education by watching our calculus lectures and spreading the word to your friends!

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[qualquan]

contd...and not for the tangent at a single point. However if we use -h our answer would have a nonzero -h addend. This too would be the answer to the secant in the opposite side of f(x). So the actual TANGENT at point f(x) will not have an h in the answer because nonzero +h and nonzero -h cancel. The second explanation is that when we use a (nonzero) h we get an answer which has a "secant error" entirely contained in the h term(s). Remaining term(s), impervious to h variation have no error.

[FreeAcademy]

@qualquan  Well this video is really to help basic calculus students, so I try to stick with established mathematics. Your debate is more suited for a number theory type course which rigorously analyses details like whether x/x cancels even if x=0. Calculus has been well established for 400 or so years so I have confident in these derivations, but if your not convinced that is certainly fine. I just don't think this is the proper forum for your concerns given the focus of this course. Thanks :)

[qualquan]

A double game is played with the value of h. We must agree that h in the denominator cannot be zero. Yet in the final quotient it is made zero to get an exact answer. So whicht is it? Is h nonzero or zero? It cannot be both just to suit our fancy. There has to be consistency otherwise it is intellectually unsatisfying. I have my own explanation where h remains nonzero all the time yet we get an exact answer such as derivative of x^2 = 2x (exact). Not enough space to elaborate unfortunately. 

[FreeAcademy]

@qualquan It is logically valid in the same sense that (x^2)/x = x even evaluated at x=0. The rules of calculus are definitely rigorous and valid, but if you don't find this derivation satisfying I have an alternative for you: infinitesimal calculus. Infinitesimal calculus is a redirection of calculus using the hyperreal number line, which is the standard number line including infinitesimals. I think it is a better approach and more physically insightful, but I taught it this way here because..

[FreeAcademy]

@qualquan ..continued.. I taught it this way because this is what many students encounter and my goal of the video series is to help students learn calculus. If you would like to learn more about infinitesimal calculus the following link is to a free ebook on the subject from the University of Wisconsin - Madison. Hope this all is a satisfying answer for you :)

math.wisc.edu/~keisler/foundat­ions.pdf

[mg2og]

@FreeAcademy So the derivative is basically the slope of the tangent

[FreeAcademy]

@mg2og Exactly!

[jkwesiga]

I like the way you present the topic. It is a lot easier to grasp with the animations and the simple example. Simple and effective! :)

[at7667a]

OMG WHOEVER INVENTED THIS IS FREAKIN AMAAZING!!! best idea i've seen in a while!

[Homehous]

Is this derivative in any way associated with the financial derivatives that has caused so much trouble in Our financial markets? And as i suspect , if that is the case , I would say that. I guess they finally found a way to use derivatives. And how is that going for the country as a whole. Maybe we should leave financial math as solid and common sense as possible.

[qualquan]

My only concern is that the traditional handling of 'h' would be confusing, particularly to the average student who would truly believe that it is zero when so summarily cancelled at the very end as if it is really zero, when it is not, instead representing a genuine "secant error". Yes it can be cancelled to get the exact solution of the tangent. The cancellation of h should be explained more rationally and hence less confusingly to the AVERAGE student. Hence many never get the real feel.

[qualquan]

I am sorry but x^2/x when x=0 is 0/0 which can be anything and not 1. So that does not help. The answer to whether h is zero or nonzero should not require a textbook. Obviously h CANNOT be zero otherwise f(x+h) would be f(x+0)=f(x). So we just cannot wish h away by making it zero in the final answer to get the correct and exact result. There are 2 logical approaches. First admit that h is nonzero and that our answer really has the nonzero h addend if we use +h. This answer is for the secant...

[suneelj93]

CRESTWOOD!

[enlargemedia]

So for the slope formula is that like the average rate of change? And the limit formula is for derivatives also known as instantaneous rate of change?