How does F'(X)=lim c->0 (F(x+c)-F(X))/c gives a derivative

[thharrimw]

I'm just learning how to use derivatives and I was wondering if anyone could explain to me how F'(X)=lim c->0 (F(x+c)-F(X))/c gives you a derivative ,what exactly is a derivative, what dose the derivative of a equation give you, and how can they be usefull in other mathmatics/ sciences.

 

[HallsofIvy]

First this question has nothing to do with "differential equations" so I am moving it to "Calculus and Analysis" where it might get more response.

As to why that limit gives the derivative, what is the precise wording of the definition of "derivative" in your textbook? In some books that formula is the definition! In other books, it is "the slope of the tangent line" and then it requires some geometry to derive that limit.

In any case, the simplest way to express why it is of use, is to point out that the derivative gives you the tangent line to the curve at a point. That is a linear function that can often be used to replace a much more complicated function.

 

[sutupidmath]

I'm just learning how to use derivatives and I was wondering if anyone could explain to me how F'(X)=lim c->0 (F(x+c)-F(X))/c gives you a derivative ,.

This is just another limit, like many others,however because it comes up a lot in many situations, and it has a great importance it has been studied more and its properties has been analyzed more, so this type of limit is called a derivative. In other words, that limit does not give you the derivative, but instead that limit by definition is called the derivative of a function!

 

[CompuChip]

Intuitively, you can think of the derivative as indicating how "steep" the function is.

Suppose you have a function F(x) and you pick some particular x. Then you can graph the function and draw the tangent line at x to that function. This tangent line tells you how steep the function is (if you zoom in far enough, the function and the tangent line will coincide). Now you can calculate the slope of the tangent line just as you would with any line, just calculate a vertical difference and divide it by the horizontal difference:
\operatorname{[Slope\ of\ F]}(x) = \frac{\Delta y}{\Delta x} = \frac{F(x + \Delta x) - F(x)}{(x + \Delta x) - x}
Draw a picture and try to identify the components of that equation (I can't do it on this forum, unfortunately).
Of course, it doesn't matter how big \Delta x is. The point is just, that as you make it smaller, the function looks more and more like the tangent line (that is, as you "zoom in" on the function it looks more and more like a straight line, which is the tangent line by construction, if you will) and therefore, the smaller you make \Delta x, the better the slope of the function (which is what the function above gives) and the slope of the tangent line (which is what you want to calculate) match. Mathematically, this means that we take the limit of \Delta x \to 0 in the above expression.

Derivatives, as the slope of a function, are an indication of how fast the function changes. If it is locally constant (like at the top of a parabola) then the derivative is zero. If the function is climbing at that point, the derivative is positive (and it's value tells you how steeply it climbs), if it is descending, the derivative is negative. This is very important information and you will find it anywhere. Most notably, in physics, if you have a function x(t) that describes the position of something as a function of time, the derivative will give you the velocity ("how fast the position changes") and the derivative of the velocity will give you the acceleration ("how fast the velocity changes").

I hope that made sense to you (I can't draw pictures here, that's too bad) -- try to draw some pictures yourself, hopefully that will make it clear what I am talking about.

 

[thharrimw]

so the derivative is the equation the tangent line of the function at a given point.

 

[Pere Callahan]

No, only the slope of that tangent line. If your tangent line to the graph of the function f at the point (x0,y0) is given by the linear equation y=mx+b then m is exactly the derivative of your function at x0, m = f'(x0)

 

[thharrimw]

so how does knowing the tangent line to a point on a equation help you?

 

[sutupidmath]

so how does knowing the tangent line to a point on a equation help you?

I am supposing that u are assuming about knowing the equation of the tangent line at a point on the curve of a function, becasue what u wrote doesn't really make sens. If you know the eq of the tangent line, then from that eq you may be able to determine its slope, which actually means that you have found the derivative of that function at that certain point.

 

[awvvu]

so how does knowing the tangent line to a point on a equation help you?

In other words, what are the applications of the derivative?

Here's some:

http://en.wikipedia.org/wiki/Differential_calculus#Applications_of_derivatives

Or, google for applications of derivatives.