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if you draw a straight line so that it touches the curve (touches, does not cross) , then the gradient of the line (y dif / x dif) is the value of the derivative at that value of x (drop a perpendicular to the x axis)

any maximum or minimum point of the graph has derivative zero at that x value.

Remember that a derivative is rate of change (or slope) of a function. If you are given an unknown curve, you can try to graph the derivative by finding strategic points. For example, if you see any minimums or maximums, you know the slope there is special (is 0) so on the graph of the derivative, those points occur at y=0. For other points that are not extrema, you can graph the slope. Eventually you should get a curve. Remember that numerically, the derivative of a polynomial is a polynomial but of a different degree so the function will look different. If it is an exponential function (like e^x) it is often the same.

Good Luck!

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At each place on the original function, you will need to estimate the slope. If the slope is +2, then that's the value of the derivative. If the slope is -3, then that's the value of the derivative.