Question

If the gradient function for f(x)=x^3+2x is g(x)=x3+2x Use algebra to show that as h→0 the gradient function approaches 3x^2+2?

Answer 1

`"see explanation"`

Explanation:

`"using differentiation from "color(blue)"first principles"`

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

`"here "f(x)=g(x)=x^3+2x`

`=lim_(hto0)((x+h)^3+2(x+h)-(x^3+2x))/h`

`=lim_(hto0)(cancel(x^3)+3x^2h+3xh^2+h^3cancel(+2x)+2hcancel(-x^3)cancel(-2x))/h`

`=lim_(hto0)(cancel(h)(3x^2+3xh+h^2+2))/cancel(h)`

`=3x^2+2`