How do you use the limit definition to find the derivative of #f(x)=2x^2+1#? Calculus Derivatives Limit Definition of Derivative 1 Answer Steve M Nov 2, 2016 # f'(x) = 4x# Explanation: By definition, # f'(x)= lim_(h->0)(f(x+h)-f(x))/h# so for # f(x) = 2x^2 +1 # we have: # f'(x) = lim_(h->0)( { (2(x+h)^2 +1) - (2x^2 +1)} )/h # # f'(x) = lim_(h->0)( (2(x^2+2hx+h^2) +1 ) - 2x^2+1)/h# # f'(x) = lim_(h->0)( 2x^2+4hx+2h^2 +1 - 2x^2-1 )/h# # f'(x) = lim_(h->0)( 4hx+2h^2 )/h# # f'(x) = lim_(h->0)( 4x+2h)# # f'(x) = 4x# 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 13574 views around the world You can reuse this answer Creative Commons License