Question

How do I find the derivative of `f(x)=x^3` from first principles?

Answer 1

First Principles `->` Difference Quotient

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

`f(x)=x^3`

`f(x+h)=(x+h)^3`

`f'(x)=lim_(h->0)((x+h)^3-x^3)/h`

`f'(x)=lim_(h->0)((x+h)(x^2+2xh+h^2)-x^3)/h`

`f'(x)=lim_(h->0)(x^3+2x^2h+xh^2+x^2h+2xh^2+h^3-x^3)/h`

`f'(x)=lim_(h->0)(2x^2h+xh^2+x^2h+2xh^2+h^3)/h`

`f'(x)=lim_(h->0)(h*(2x^2+xh+x^2+2xh+h^2))/h`

`f'(x)=lim_(h->0)2x^2+xh+x^2+2xh+h^2`

`f'(x)=lim_(h->0)3x^2+xh+2xh+h^2`

`f'(x)=3x^2+x(0)+2x(0)+(0)^2`

`f'(x)=3x^2`