Q6 - Calculus

Question

Why does the proof of the power rule (Theorem 2.9) in this text work only when k is an integer? Also, at which point in the proof do we use the fact that k is positive?

Step-by-Step Solution

Answer

$\underset{z\to x}{\lim }\frac{z^k-x^k}{z-x}$

If k is other than positive then division is not possible. 

Step 1: Given Information

the proof of the power rule (Theorem 2.9) in this text work only when k is an integer 

Step 2: Derivative power rule

Lets consider an expression 

$$\begin{gathered} \underset{z\to x}{\lim }\frac{z^k-x^k}{z-x} \\ we need to prove \\ \underset{z\to x}{\lim }\frac{z^k-x^k}{z-x}=k{x}^{k-1} \end{gathered}$$

k is positive  because $z^k-x^{k }$can be expressed as

$z^k-x^{k }=(z-k)(z^{k-1}+z^{k-2}x+z^{k-3}{x}^2.......z^2{x}^{k-3}+z^1{x}^{k-2}+ x^{k-1}$

If k is other than positive then division is not possible.