How do I sketch the graph of the derivative of a function?

[aero]

I'm having a hard time sketching the graphs of the derivatives of unknown functions. I can't do it if I'm looking at just a graph of some unknown function. Can someone please explain in a clear way how to approximate and sketch the graph of the derivative?

[Jones]

From the original graph, the maxiumim and minimum points are where the derivative crosses the x-axis

Then, from the original function, wherever there is a POSITIVE gradient, the line is ABOVE the x-axis

And inversely, wherever there is a NEGATIVE gradient, the line is BELOW the x-axis.

This video explains it more detail

[Millan]

So, the first thing you want to do is graph the points where it crosses the x-axis. This is going to be where the slope of the original equals zero (this makes sense right, because derivative is just slope and the slope = zero there). This is at the extrema (max and min). Then, you estimate the slopes of the other parts of the graph using the average slope formula ( f(b) - f(a) ) divided by (b - a), where b and a are x-values that you chose to approximate slope for. Sketch in a nice fit ^_^

A good way to check to see that you did it right is to look at what type of function the derivative graph is (linear, quadratic, etc.). It should always be of degree ONE LESS than the original, meaning the highest power is one less than that of the original function. Hope this helps!