Question

Use the$h\to 0$definition of the derivative to prove the power rule holds for positive integers powers

Short answer

We prove the power rule holds for positive integers powers using the definition of derivative

Step-by-step solution

Given information

We are given a function as$f\left(x\right)=x^n$, Where n is a positive integer

Find the derivative

We use definition of the derivative

We have,

$$\begin{gathered} \underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h} \\ \underset{h\to 0}{\lim }\frac{{(x+h)}^n-x^n}{h} \\ \underset{h\to 0}{\lim }\frac{x^n+n{x}^{n-1}h+.....+h^n-x^n}{h}[binomialexpansion] \\ \underset{h\to 0}{\lim }\frac{n{x}^{n-1}h+.....+h^n}{h} \\ \underset{h\to 0}{\lim }\frac{h(n{x}^{n-1}+....+h^{n-1)}}{h} \\ \underset{h\to 0}{\lim }(n{x}^{n-1}+....+h^{n-1)} \\ =n{x}^{n-1} \end{gathered}$$