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 xn – 1h + nC2 xn – 2h2 + …… + nCn hn
Take h common –
As there is no more indeterminate, so put value of h to get the limit.
⇒ f’(x) = k nC1 xn – 1 = k nxn – 1
Hence,
Derivative of f(x) = kxn is k nxn – 1