Question

Find the derivative of f(x) = xⁿ, where n is positive integer, by first principle.

Answer 1

By definition,

f'(x) = ${\displaystyle \frac{f(x+h)-f\left(x\right)}{h}}$

= ${\displaystyle \frac{{(x+h)}^n-x^n}{h}}$

Using Binomial theorem,

We have ${\displaystyle {(x+h)}^n={}^n{C}_0{x}^n+{}^n{C}_1{x}^{n-1}h+...+{}^n{C}_n{h}^n}$

Thus, f'(x) = ${\displaystyle \underset{h\to 0}{lim}\frac{{(x+h)}^n-x^n}{h}}$

= ${\displaystyle \underset{h\to 0}{lim}\frac{h(n{x}^{n-1}+...+h^{n-1})}{h}}$

= ${\displaystyle n{x}^{n-1}}$