Using the limit definition, how do you differentiate #f(x)=x^2+3x+1#? Calculus Derivatives Limit Definition of Derivative 1 Answer Mr. Mike Apr 11, 2018 #f'(x)=2x+3# Explanation: #f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h# In this case #lim_(hrarr0)(f(x+h)-f(x))/h# #=lim_(hrarr0)((x+h)^2+3(x+h)+1-x^2-3x-1)/h# #=lim_(hrarr0)(x^2+2xh+h^2+3x+3h+1-x^2-3x-1)/h# #=lim_(hrarr0)(2xh+h^2+3h)/h# #=lim_(hrarr0)2x+h+3=2x+3#. 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 7714 views around the world You can reuse this answer Creative Commons License