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I've been reading up on functions and I'm not sure I quite understand it all. From what I understand if you have a function f(x)=x^2 then if you input x as 2 the output, y, or codomain will be 4. The domain is the input. Now in order to graph this function you simply input a bunc h of numbers such as 2, 4, 6, 8, and 10 and the output or range will be 4, 16, 36, 64, and 100. If we graph these numbers we can find the derivative or slope of the function. Is all of this correct?
Most of what you say is correct, except that the derivative is not determined by measuring the slope of the graph.
A function takes one or more input values each within some domain, and returns an output value in another domain, called the range. For f(x)=x^2 both the domain of x, and the range of f are the real number (all numbers from negative to positive infinity)
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Note: the following is a small introduction to calculus.
To measure/calculate the slope, you take two points, let's call them (a,f(a)) and (b,f(b)), their distance in one direction is Δx=a-b, and in the other Δy=f(a)-f(b). The slope between these two points on the graph is Δy/Δx.
The derivative is the slope in a single point in the grave, i.e. Δx=0. If you were to calculate this, it would lead to a devision by zero, therefore you need a trick.
We denote the slope of the function f at a point x as f'(x), and we use h to denote a very small value (serving the function of Δx above)
f'(x)= ( f(x) - f(x+h) ) / (x-(x+h))
= (x^2 - (x+h)^2)/h
= (x^2 - x^2 + 2hx - h^2)/h
= 2x - h
Now, you can make h=0 and get
f'(x)=2x
You can use f'(x) to get the slope of f(x) in any point x. f'(x) is called the derivative of f(x) another common notation is:
That notation is very similair to the Δy/Δx notation used for slopes between two points.Code:dy
-- f(x)
dx
There's lots of notations for it, but thats not one of them, either drop f(x) or y out of it (they're the same thing)
other notations are:
f'(x)
y'
Df(x)
Dy (sometimes with an index x after D)
The derivative (slope) is defined as the limit of the function twanvl demonstrated as follows: f'(x)= lim h->0 (f(x+h)-f(x) ) / h. (btw the other way around there in the numerator)
The limit doesn't always exist, the function has to be contigous (without jumps) at that point and its left and right limits must exist and coincide. look at the notation here: http://mathworld.wolfram.com/Limit.html
usually when you calculate derivates you never use the definition but use some basic rules derived from it, but its a good to know the idea behind it anyway.