Chapter 5.2, Problem 19ES

(For students who have Studied calculus) Use mathematical induction, the Product rule from calculus, and the facts that $\frac{d\left(x\right)}{dx}=1$ and that $x^{k+1}=x\cdot x^k$ to prove that for every integer $n\ge 1$ , $\frac{d\left(x^n\right)}{dx}=n{x}^{n-1}$ .