Chapter 3.5, Problem 92E

Proof

(a) Prove (Theorem 3.3) that $d/dx[x^n]=n{x}^{n-1}$ for the case in which r is a rational number. (Hint: Write $y=x^{p/q}$ in the form $y^{″}=x^n$ and differentiate implicitly. Assume that p and u are integers, where q > 0.)

(b) Prove part (a) for the case in which r is an irrational number. (Hint: Let $y=x^r$ where r is a real number, and use logarithmic differentiation.)