Problem

Let

Find g'(x), g''(x), and g'''(x), and graph them. Can you find a formula for the nth derivative of g(x)?

Solution

g(x) can be written

We can use the power rule to find its derivatives.

To find a formula for the nth derivative, we notice that the power of x in the denominator is always one more than the order of the derivative. Also, the sign of the result alternatives between positive and negative, with odd derivatives being negative and even derivatives positive. Finally, the coefficient or numerator is equal to the product of integers up to the order of the derivative.