Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 82

Determine the $n+h$ derivatives of each function by calculating the first few derivatives
and observing the pattern that occurs.

a.) $f(x) = x^a$
b.) $\displaystyle f(x) = \frac{1}{x}$

a.) Using Power Rule,

$\begin{equation} \begin{aligned} f(x) &= x^a\\ f'(x) &= ax^{a-1}\\ f''(x) &= a(a-1) x^{a-2}\\ f'''(x) &= a(a-1)(a-2)x^{a-3} \end{aligned} \end{equation}$


From these derivatives, we can make the pattern and obtain the formula for the $n+h$ derivatives as...
$f^{(n)}(x) = a(a-1)(a-2) \cdots (a-n+1) x^{a-n}$

b.) We can rewrite $\displaystyle f(x) = \frac{1}{x}$ as $f(x) = x^{-1}$ to make it easier to differentiate.


$\begin{equation} \begin{aligned} f(x) &= x^{-1}\\ f'(x) &= -1x^{-2}\\ f''(x) &= -1(-2)x^{-3}\\ f'''(x) &= -1(-2)(-3)x^{-4} \end{aligned} \end{equation}$

By these derivatives, we can make out the pattern and formulate the $n+h$ derivatives as...

$f^{(n)} (x) = (-1)^{(n)} n! x^{-(n+1)}\\ \displaystyle f^{(n)} (x) = (-1)^n \frac{n!}{x^{-(n+1)}}$