Question

Differentiate y=x^n using first principle?

Answer 1

`(dy)/(dx)=nx^(n-1)`

Explanation:

According to first principle for `y=f(x)`

`(dy)/(dx)=lim_(h->0)(f(x+h)-f(x))/h`

as `y=f(x)=x^n`, we have `f(x+h)=(x+h)^n`

and `(dy)/(dx)=lim_(h->0)((x+h)^n-x^n)/h`

and using Binomial theorem this is

`lim_(h->0)(x^n+nhx^(n-1)+(n(n-1))/(2!)h^2x^(n-2)+(n(n-1)(n-2))/(3!)h^3x^(n-2)+........-x^n)/h`

= `lim_(h->0)(nhx^(n-1)+(n(n-1))/2h^2x^(n-2)+(n(n-1)(n-2))/6h^3x^(n-2)+........)/h`

= `lim_(h->0)(nx^(n-1)+(n(n-1))/2hx^(n-2)+(n(n-1)(n-2))/6h^2x^(n-2)+........)`

= `nx^(n-1)`