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Consider the following functions. 
\begin{enumerate}
  \item	$ f(x) = \sin(x)$ 
  \item	$f(x) = (1/x)$ 
  \item   $f(x) = \sqrt{x}$ 
  \end{enumerate}

There is a pattern in the derivatives of these function. We will discover and establish it.

  \begin{enumerate}  
    \item[(a)] Compute the first, second, and third derivative of each of these functions. 
    \item[(b)] Propose a general formula for $f^{(n)}(x)$. This will be in terms of $n$.
    \item[(c)] Verify the formula using a computer system. Check what the 10th derivative of the function is from your formula, then plug that into a computer system (Wolfram Alpha, for example) to see what it gives for the 10th derivative. They should match!
    \item[(d)] \textbf{Bonus. Do not turn in.}Verify the proposed formula in two steps. Step $1$ - verify the formula when $n=1$. Step $2$- assuming formula is true for a certain value of $n$, let us say for $n=k$, show that formula is valid for the next value of $n=k+1$.
	 \end{enumerate}

This last part is an elegant way of establishing the truth of statements like `` for all $n$, something
statement involving  $n$ is valid ''. Convince yourself it works! This method is called ' mathematical
induction'. 

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