Using the limit definition, how do you differentiate #f(x)=x^3−7x+5#? Calculus Derivatives Limit Definition of Derivative 1 Answer Harish Chandra Rajpoot Jul 3, 2018 #f'(x)=3x^2-7# Explanation: Given function: #f(x)=x^3-7x+5# #f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}# #=\lim_{h\to 0}\frac{(x+h)^3-7(x+h)+5-(x^3-7x+5)}{h}# #=\lim_{h\to 0}\frac{(x+h)^3-x^3-7h}{h}# #=\lim_{h\to 0}\frac{(x+h-x)((x+h)^2+x^2+(x+h)x)-7h}{h}# #=\lim_{h\to 0}\frac{h((x+h)^2+2x^2+hx)-7h}{h}# #=\lim_{h\to 0}((x+h)^2+2x^2+hx-7)# #=((x+0)^2+2x^2+0\cdot x-7)# #=3x^2-7# 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 1516 views around the world You can reuse this answer Creative Commons License