Outside of basic calculus, is the power rule viable to find a derivative?

Answer 1

Yes. The Power Rule in differentiation continues throughout until Advanced Calculus. The Power Rule is also broken down and defined as a sequence proven in advanced mathematics; specifically the course of Real Analysis.

Proof: We prove the Power Rule viable to finding a derivative with positive integers n greater than or equal to 1 by using the binomial theorem plus definition of a derivative such that an exponential function xⁿ possesses a derivative:d/dx (xⁿ) = lim h->0 [f(x+h) - f(x)] / h = [(x+h)ⁿ-xⁿ] / h and the binomial theorem (x+h)ⁿ = xⁿ+nx^n-1*h+[n(n-1)/2]*x^n-2h²+***+nxh^n-1+hⁿ. We then replace (x+h)ⁿ with the sequence defined to obtain the derivative,

lim h->0 [xⁿ+nx^n-1*h+[n(n-1)/2]*x^n-2h²+***+nxh^n-1+hⁿ] - xⁿ all over h

= [nx^n-1*h] / h = nx^n-1.

(End Proof)