By now you are comfortable with the graphical implications of the first and second derivatives of a function. $f^\prime$ gives you the slope of the tangent line at a point while $f^{\prime\prime}$ 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. $f^{\prime\prime\prime}$ is simply the rate of change of $f^{\prime\prime}$ and so on.

1. What is $f^{(4)}(x)$ if $f(x) = x^8$?

   Hint (a):

   Take the derivative four times.

2. If $f(x) = e^{2x}$, write an expression for the $n^{\text{th}}$ derivative, $f^{(n)}(x)$.

   Hint (b):

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