How do you find f'(x) using the definition of a derivative for #f(x)=(1-6t)/(5+t)#? Calculus Derivatives Limit Definition of Derivative 1 Answer Dharma R. Sep 23, 2015 #-31/(5+t)^2# Explanation: #f(t)=(1-6t)/(5+t)# #f(t+h)=(1+6(t+h))/(5+(t+h))# #f(t+h)-f(t)=(1-6(t+h))/(5+(t+h))-(1-6t)/(5+t)=(5+t-6(t^2+5h+5t+ht)-5-t-h+30t+6t^2+6ht)/((5+t+h)(5+t))=(-31h)/((5+t+h)(5+t))# #Lt_(h->0)((f(t+h)-f(h))/h)=Lt_(h->0)(-31h)/(h(5+t+h)(5+t))=Lt_(h->0)(-31)/((5+t+h)(5+t))=-31/(5+t)^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 1742 views around the world You can reuse this answer Creative Commons License