What is the equation of the tangent line of #f(x)=xlnx^2-x^2lnx# at #x=3#? Calculus Derivatives Tangent Line to a Curve 1 Answer Bio Nov 6, 2015 #y=-[1+4ln(3)]x+3[1+3ln(3)]# Explanation: #f(x)=xln(x^2)-x^2ln(x)# #=(2-x)xln(x)# The equation of the tangent line #x=3# is the same as the first order taylor expansion of #f# about #x=3#. #f(x)~~f(3)+f'(3)(x-3)=y# #f(3)=(2-3)(3)ln(3)# #=-3ln(3)# #f'(x)=frac{d}{dx}((2-x)xln(x))# #=(2-x)xfrac{d}{dx}(ln(x))+(2-x)frac{d}{dx}(x)ln(x)+frac{d}{dx}(2-x)xln(x)# #=(2-x)xfrac{1}{x}+(2-x)(1)ln(x)+(-1)xln(x)# #=2-x+2(1-x)ln(x)# #f'(3)=2-3+2(1-3)ln(3)# #=-1-4ln(3)# #y=-(1+4ln(3))(x-3)-3ln(3)# #=-(1+4ln(3))x+9ln(3)+3# Answer link Related questions How do you find the equation of a tangent line to a curve? How do you find the slope of the tangent line to a curve at a point? How do you find the tangent line to the curve #y=x^3-9x# at the point where #x=1#? How do you know if a line is tangent to a curve? How do you show a line is a tangent to a curve? How do you find the Tangent line to a curve by implicit differentiation? What is the slope of a line tangent to the curve #3y^2-2x^2=1#? How does tangent slope relate to the slope of a line? What is the slope of a horizontal tangent line? How do you find the slope of a tangent line using secant lines? See all questions in Tangent Line to a Curve Impact of this question 1314 views around the world You can reuse this answer Creative Commons License