today my lecturer taught us derivatives. So I am a complete newbie at this and I am struggling with the graphic of a derivative. To be more specific, I want to know how we can "read" a graphic of a derivative and conclude some stuff about the given function?
Example:
What we can "read" from the blue graph about the red one and how are they connected with each other?
I know this might be a stupid question but I honestly can't get it.
Best Answer
The derivative of a function gives the slope of the tangent line to that function at the same $x$ co-ordinate, so looking at a graph of the derivative tells you something about how the function is moving at various points. In particular:
If the derivative is 0, then the tangent line is horizontal, so the function must go completely flat at that point (either a local maximum or minimum, or a point of inflexion). For example, on your graph, the derivative is zero at -1, 0 and 1, which are all points where the function itself is completely flat.
If the derivative is positive, then the tangent line is diagonal going up-right, so the function is increasing at that point. For example, the derivative in your graph is positive between -1 and 0, and from 1 onwards, and that's where the graph slopes up.
If the derivative is negative, then the tangent line is diagonal down-right, so the function is decreasing at that point.
If you want to get fancier, then since bigger values of the derivative (either positive or negative) mean more significant slopes, then the fact that your derivative trends to infinity in both directions means that your graph will get more extremely sloping as it goes along. By comparison, if the derivative stayed very small, then the function would be much more gentle even as it trended wherever it was going.