Question

How do you use the definition of a derivative to find the derivative of `f(x)= 2x^2-x`?

Answer 1

`f'(x)=4x-1`

Explanation:

`"differentiating from "color(blue)"first principles"`

`f'(x)=lim_(hto0)(f(x+h)-f(x))/h`

`color(white)(f'(x))=lim_(hto0)(2(x+h)^2-(x+h)-(2x^2-x))/h`

`color(white)(f'(x))=lim_(hto0)(2x^2+4hx+2h^2-x-h-2x^2+x)/h`

`color(white)(f'(x))=lim_(hto0)(4hx+2h^2-h)/h`

`color(white)(f'(x))=lim_(hto0)(cancel(h)(4x+2h-1))/cancel(h)`

`color(white)(xxxx)=4x-1`