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

1 Answer
Steve M
Oct 24, 2016

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

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