How to know the derivative of a function by looking at the graphs?

The derivative is the gradient.

At any local Max's or min's the derivative graph will cut the x axis.

For example a graph x^2
The derivative will have a positive gradient as the gradient is increasing at the lower values then at x=0 the gradient is 0 so the derivative graph will pass the point (0,0). Remember that the derivative graph will be linear.

To get more detail find the points the graph crosses the x axis and put into for a(x-q)(x-p)=0 you will have to solve for 'a' by finding a point on the graph and substituting it in. Then you can find the derivative of that function and graph it

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