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By now you are comfortable with the graphical implications of the first and second derivatives of a function. gives you the slope of the tangent line at a point while tells you the concavity of the graph at a point. There is nothing to stop us from finding the third and fourth (and beyond!) derivatives, although there are not any significant graphical characteristics associated with higher derivatives. is simply the rate of change of and so on.

  1. What is if ?

    Take the derivative four times.

  2. If , write an expression for the derivative, .

    Take a few derivatives of this function and look for patterns.