Question

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

Answer 1

By definition,

f'(x) = `(f(x + h) - f(x))/h`

= `((x + h)^n - x^n)/h`

Using Binomial theorem,

We have `(x + h)^n = ""^nC_0 x^n + ""^nC_1 x^(n - 1) h + ... + ""^nC_n h^n`

Thus, f'(x) = `lim_(h -> 0) ((x + h)^n - x^n)/h`

= `lim_(h -> 0) (h(nx^(n - 1) + ... + h^(n - 1)))/h`

= `nx^(n - 1)`