Chapter 3.2, Problem 64E

(a) If g is differentiable, the Reciprocal Rule says that

$\frac{d}{dx}\left[\frac{1}{g(x)}\right]=-\frac{g\text{'}(x)}{{[g(x)]}^2}$

Use the Quotient Rule to prove the Reciprocal Rule.

(b) Use the Reciprocal Rule to differentiate the function in Exercise 16.

(c) Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is,

$\frac{d}{dx}(x^{-n})=-n{x}^{-n-1}$

for all positive integers n.