derivative, the way to show in a graph

Question

http://www.wolframalpha.com/input/?i=derivative+x%5E2%2C+x%5E2

Just reminding myself some math..

Is it ok to show the derivative in such a way like was shown in the link above for $x^2$ function?

A derivative is supposed to be for some point of function. So, then I should see lots of lines, but not one. But if one then what is the 'point' where it is applied for (for that function is depicted) ?

Answer 1

You are confusing "derivative" and "tangent line". A derivative is a function which keeps track of the slopes of tangent lines.

At each point the derivative is a number (the slope of a the corresponding tangent at that point) whereas the tangent line is determined by an equation (which keeps track of not only the slope but also the point of tangency).

To incorporate kaine's comment: You can get Wolfram Alpha to draw a specific tangent line at a point using...

tangent line of SOMETHING at x=SOME NUMBER

For example:

tangent line of x^2 at x=0.5

Answer 2

A derivative is the slope of the tangent at a point. I think you are thinking of tangents or secants. (You can approximate the average rate of change using secants.)

Since the derivative of $x^{2}$ is $2x$, Wolfram Alpha just plotted both $f(x)=x^2$ and $f'(x)=2x$.