How do you use the limit definition to find the derivative of #f(x)=x^3+1#? Calculus Derivatives Limit Definition of Derivative 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 # Answer link Related questions What is the limit definition of the derivative of the function #y=f(x)# ? Ho do I use the limit definition of derivative to find #f'(x)# for #f(x)=3x^2+x# ? How do I use the limit definition of derivative to find #f'(x)# for #f(x)=sqrt(x+3)# ? How do I use the limit definition of derivative to find #f'(x)# for #f(x)=1/(1-x)# ? How do I use the limit definition of derivative to find #f'(x)# for #f(x)=x^3-2# ? How do I use the limit definition of derivative to find #f'(x)# for #f(x)=1/sqrt(x)# ? How do I use the limit definition of derivative to find #f'(x)# for #f(x)=5x-9x^2# ? How do I use the limit definition of derivative to find #f'(x)# for #f(x)=sqrt(2+6x)# ? How do I use the limit definition of derivative to find #f'(x)# for #f(x)=mx+b# ? How do I use the limit definition of derivative to find #f'(x)# for #f(x)=c# ? See all questions in Limit Definition of Derivative Impact of this question 2758 views around the world You can reuse this answer Creative Commons License