What is the relation between the graph of a function and the graph of it's derivative? Suppose that the function is x^2. It's derivative is 2x. If we graph both x^2 and 2x on the same coordinate axes, can we conclude anything about x^2 by looking at the graph of 2x, or vice versa (i.e. can we conclude anything about 2x by looking at the graph of x^2)? For example, can we tell what will be the slope of the tangent line at a curve at x^2 by looking at the graph of it's derivative 2x? If there is no relation between the two, then what is the use of graphing the derivative of a function?