Question

In the subexample of the quotient rule, I got d/dx (1/(x^n)) = -n^(-1)x^(-n+1). How did the substitute teacher get d/dx (1/(x^n)) = (-n) x^(-1-n)?

Answer 1

\[d/dx (x^{n}) = [x^n *d/dx(1) - 1*d/dx(x^n)]/(x^n)^2\]
\[= [0-nx^{n-1}]/x^{2n} = -nx^{n-1}*x^{-2n}=-nx^{n-1-2n}=-nx^{-1-n}\]

Answer 2

there is an error, you may write:
\[d/dx (1/x^{n}) = [x^n *d/dx(1) - 1*d/dx(x^n)]/(x^n)^2\]