How do you find the derivative of # 1/(x^2-1)# using the limit definition? Calculus Derivatives Limit Definition of Derivative 1 Answer Shwetank Mauria Sep 19, 2016 #(df)/(dx)=(-2x)/(x^2-1)^2# Explanation: as #f(x)=1/(x^2-1)# #f(x+h)=1/((x+h)^2-1)# Hence, #f(x+h)-f(x)=1/((x+h)^2-1)-1/(x^2-1)# = #((x^2-1)-((x+h)^2-1))/((x^2-1)((x+h)^2-1))# = #((x^2-1)-(x^2+2hx+h^2-1))/((x^2-1)((x+h)^2-1))# = #((x^2-1-x^2-2hx-h^2+1))/((x^2-1)((x+h)^2-1))# = #((-2hx-h^2))/((x^2-1)((x+h)^2-1))# and #(f(x+h)-f(x))/h=((-2x-h))/((x^2-1)((x+h)^2-1))# Now #(df)/(dx)=Lt_(h->0)(f(x+h)-f(x))/h# = #Lt_(h->0)((-2x-h))/((x^2-1)((x+h)^2-1))# = #(-2x)/((x^2-1)(x^2-1))# = #(-2x)/(x^2-1)^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 10817 views around the world You can reuse this answer Creative Commons License