Find the derivative of #x+sqrtx# using the definition of derivative? Calculus Derivatives First Principles Example 1: x² 1 Answer Shwetank Mauria Apr 14, 2017 #d/(dx)(x+sqrtx)=1+1/(2sqrtx)# Explanation: #(df)/(dx)# is defined as #Lt_(h->0)(f(x+h)-f(x))/h# Here #f(x)=x+sqrtx# and hence #f(x+h)=x+h+sqrt(x+h)# and #f(x+h)-f(x)=h+sqrt(x+h)-sqrtx# and hence #(df)/(dx)=Lt_(h->0)[1+(sqrt(x+h)-sqrtx)/h]# = #1+Lt_(h->0)((sqrt(x+h)-sqrtx)(sqrt(x+h)+sqrtx))/(h(sqrt(x+h)+sqrtx))# = #1+Lt_(h->0)h/(h(sqrt(x+h)+sqrtx))# = #1+Lt_(h->0)1/(sqrt(x+h)+sqrtx)# = #1+1/(2sqrtx)# Answer link Related questions How you you find the derivative #f(x)=x^2# using First Principles? What is the power rule derivative? How do you differentiate #f(x) = 3#? How do you differentiate #f(x) = x^2 - 4x + 3#? Question #ae316 How do you find the derivative of #f(x) = 1/sqrt(2x-1)# by first principles? Find the derivative of #sinx# using First Principles? How would you solve this? How do we find the differential of #y=x^2+1# from first principle? Question #69fe4 See all questions in First Principles Example 1: x² Impact of this question 2100 views around the world You can reuse this answer Creative Commons License