Q. 85 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given information

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

Step 2: Find the derivative

We use definition of the derivative 

We have,

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