Question

How do you use the limit definition to find the derivative of `f(x)=x^3+1`?

Answer 1

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

Explanation:

By definition `f'(x) =lim_(hrarr0)(f(x+h)-f(x))/h`

So, with `f(x)=x^3+1` we have:

`f'(x) =lim_(hrarr0) (((x+h)^3+1 ) - (x^3+1) ) / h`
`:. f'(x) =lim_(hrarr0) (((x^3+3hx^2+3h^2x+h^3)+1 ) - x^3-1 ) / h`
`:. f'(x) =lim_(hrarr0) (x^3+3hx^2+3h^2x+h^3+1 - x^3-1 ) / h`
`:. f'(x) =lim_(hrarr0) ( 3hx^2+3h^2x+h^3 ) / h`
`:. f'(x) =lim_(hrarr0) ( 3x^2+3hx+h^2 )`
`:. f'(x) =3x^2`