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#1 2007-06-14 13:32:20

MathsIsFun
Registered: 2005-01-21
Posts: 7,685

Introduction to Derivatives

A very long-winded Introduction to Derivatives

And a shorter Derivatives as dy/dx

Could someone please check them for validity?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#2 2007-06-14 22:10:24

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Introduction to Derivatives

Nice pages! Very well explained, and you've got first principles in there as well.

One small thing: When you're finding the derivative of x², I would personally cancel Δ's on the numerator and denominator before starting to neglect things because they're close to 0.
Neglecting Δx² but not Δx seems a bit illogical if you're doing it that way around. Maybe that's just my taste.

Why did the vector cross the road?
It wanted to be normal.

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#3 2007-06-15 02:18:02

ganesh
Registered: 2005-06-28
Posts: 25,063

Re: Introduction to Derivatives

Nice pages. No errors spotted when rushed through. Derivatives of other functions from first principles can be added, and Sinx, and other trignometric functions.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#4 2007-06-16 16:37:00

MathsIsFun
Registered: 2005-01-21
Posts: 7,685

Re: Introduction to Derivatives

mathsyperson wrote:

One small thing: When you're finding the derivative of x², I would personally cancel Δ's on the numerator and denominator before starting to neglect things because they're close to 0....

I agree ... have updated. What do you think?

And thanks, ganesh, will try to add another page with lots of "derivations of derivatives" ... using dx notation?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#5 2007-06-18 15:12:11

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

Re: Introduction to Derivatives

Pretty nice!!
Now, just food for thought, think about 0/0 being about 1, or undefined.
Then look at your simplification after this quote:
"The first thing we can do is simplify it, because Δx is at both the top and bottom:"
Looks like delta x over delta x is one to me!!

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#6 2007-06-18 16:04:27

MathsIsFun
Registered: 2005-01-21
Posts: 7,685

Re: Introduction to Derivatives

Yes, it is! But remember, Δx cannot be zero

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#7 2007-06-18 16:13:46

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Introduction to Derivatives

Ah, but MathIsFun, what if Δx was 0?  Then it must be that Δf(x) is also 0, no matter what the function.  So the question we are asking is "What is the slope of a single point of any function?"  if we consider Δf(x)/Δx.  And the answer to this, which happens to be the same as the answer to 0/0, is indeterminate.  Because a function can have any slope at a single point.

Gotta love how two independant lines of thought wind up giving precisely the same answer.  Gotta love math.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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