Question #e91c7 Calculus Graphing with the Second Derivative Relationship between First and Second Derivatives of a Function 1 Answer Andrea S. Apr 10, 2017 #d/dx (lnx)^2 = (2lnx)/x# #d^2/dx^2 (lnx)^2 = (2 -2 lnx )/x^2# Explanation: Using the chain rule: #d/dx (lnx)^2 = (2lnx)( d/dx lnx )= (2lnx)/x# Then: #d^2/dx^2 (lnx)^2 = d/dx ((2lnx)/x) = 2 d/dx(lnx/x)# and using the quotient rule: #d^2/dx^2 (lnx)^2 = 2 (x d/dxlnx - lnx d/dx x)/x^2# #d^2/dx^2 (lnx)^2 = 2 (x *1/x - lnx )/x^2# #d^2/dx^2 (lnx)^2 = 2 (1 - lnx )/x^2# Answer link Related questions What is the relationship between the First and Second Derivatives of a Function? Question #64fc4 What are the first two derivatives of #y = 2sin(3x) - 5sin(6x)#? What is the second derivative of the function #f(x)=sec x#? If #f(x)=sec(x)#, how do I find #f''(π/4)#? What is the second derivative of #g(x) = sec(3x+1)#? How do you use the second derivative test to find the local maximum and minimum for... What is the first and second derivative of #1/(x^2-x+2)#? What is the second derivative of #x/(x-1)# and the first derivative of #2/x#? What does the 2nd Derivative Test tell you about the behavior of #f(x) = x^4(x-1)^3# at these... See all questions in Relationship between First and Second Derivatives of a Function Impact of this question 1254 views around the world You can reuse this answer Creative Commons License