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November 29th, 2012, 05:34 AM   #1
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Fundamental Derivative question

I know very well, How to derivate given function.
What I would love to know is Why we derivate? and what do we exactly get after differenciating?

Well, What I know is by derivating we can get slope of the tangent of the given point of graph. But then immediate next question would be, what is use of it?

lastly, the regular differenciation formula says, f'(x) = lim h->0 (f(x+h)-f(x)/h)

Why only limit of that equation is derivative and not every other limit is a derivative.

My questions are amateur I am aware of that but its the fundamental knowledge I want to have
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November 29th, 2012, 07:48 AM   #2
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Re: Fundamental Derivative question

(f(x+h)-f(x))/h gives us the slope.
Let's see how it works: We have two points, (x, y) and (x+h, y). In functions graph, we use the f(x) value for y. So, our points are these: (x, f(x)) and (x+h, f(x+h)).
The slope of the line which contains these number is m=(f(x+h)-f(x))/h.
Why we take the limit when h approaches 0?
Well, we want to know the slope at any point. The equation gives us the slope in range [x, x+h]. We have to narrow the range, so that we can find the slope. However, we can't take h is equal to true because it's the denominator. So, we take the limit when h goes to 0.
I wish it helps.
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November 30th, 2012, 04:53 AM   #3
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Re: Fundamental Derivative question

Well, Thank you for explanation but I would like to know, Why we calculate slope of tangent
What can we achieve by calculating slope of tangent.
Well, Additionally, Why do we treat only that formula as the formula of calculating derivatives and not any other limit based function.

I am sorry if my questions are too basic, I know how to 'solve' this stuff, But I am curious about WHY we do all that...

Please help me understand fundamental logic behind them.

Thank you.
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November 30th, 2012, 10:18 AM   #4
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Re: Fundamental Derivative question

It would be better to actually take a Calculus course!

Remember that Newton (and Leibniz) first developed Calculus because of a fundamental "paradox" in gravitational theory. That is, if the gravitational force on an planet depended (as Newton suspected) on the distance then, since at any given instant the planet has a specific distance from the sun, then the force has a specifi value at that given intstant. Now, F= ma so a= F/m is the acceleration at that instant. But, while we are comfortable with the idea of an acceleration or velocity at a given instant, now, at that time the only possible concepts of acceleration and velocity were "average" acceleration and speed- change in speed or distance divided by change in time. If there is no change in time, there cannot be any motion and so no speed or acceleration.

Yet, we are asserting that the sun's gravity exerts a certain force on a planet, and so gives it a specific acceleration, at a give instant. How can that be? In order to make sense of that, we must have a way of defining rate of change at a give instant. And that is the idea of the "derivative". It is the rate of change of a function, f(x), at a given value of x rather than an average value.
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