What is the first derivative and second derivative of #f(x) = (x-1)^2 (x-3)#? Calculus Basic Differentiation Rules Product Rule 1 Answer RedRobin9688 Jun 27, 2018 #f'(x)=3(x-1)^2-4(x-1)# #f''(x)=6x-10# Explanation: Let #u=x-1 , x-3=u-2 therefore (du)/dx=1# #u^2(u-2)=u^3-2u^2# #color(blue)(d/dx)(u^3-2u^2)# #3u^2*(du)/dx-4u(du)/dx# Substitute #f'(x)=3(x-1)^2-4(x-1)# #color(blue)(d/dx)(3u^2-4u)# #6u(du)/dx-4(du)/dx# Substitute #f''(x)=6(x-1)-4=6x-10# Answer link Related questions What is the Product Rule for derivatives? How do you apply the product rule repeatedly to find the derivative of #f(x) = (x - 3)(2 - 3x)(5 - x)# ? How do you use the product rule to find the derivative of #y=x^2*sin(x)# ? How do you use the product rule to differentiate #y=cos(x)*sin(x)# ? How do you apply the product rule repeatedly to find the derivative of #f(x) = (x^4 +x)*e^x*tan(x)# ? How do you use the product rule to find the derivative of #y=(x^3+2x)*e^x# ? How do you use the product rule to find the derivative of #y=sqrt(x)*cos(x)# ? How do you use the product rule to find the derivative of #y=(1/x^2-3/x^4)*(x+5x^3)# ? How do you use the product rule to find the derivative of #y=sqrt(x)*e^x# ? How do you use the product rule to find the derivative of #y=x*ln(x)# ? See all questions in Product Rule Impact of this question 2031 views around the world You can reuse this answer Creative Commons License