Differentiate each of the following from first principles:

kxⁿ

We need to find the derivative of f(x) = kxⁿ

Derivative of a function f(x) from first principle is given by –

{where h is a very small positive number}

∴ derivative of f(x) = kxⁿ is given as –

Using binomial expansion we have –

(x + h)ⁿ = ⁿC₀ xⁿ + ⁿC₁ x^{n – 1}h + ⁿC₂ x^{n – 2}h² + …… + ⁿC_n hⁿ

Take h common –

As there is no more indeterminate, so put value of h to get the limit.

⇒ f’(x) = k ⁿC₁ x^{n – 1} = k nx^{n – 1}

Hence,

Derivative of f(x) = kxⁿ is k nx^{n – 1}