How do you differentiate #f(x)= (x^3+2x+1)(2+x^-2)# using the product rule? Calculus Basic Differentiation Rules Product Rule 1 Answer Steve M Oct 7, 2016 #f'(x)=6x^2-x^-2-2x^-3-2# Explanation: The product rule is # d/dx(uv)=u (dv)/dx+v (du)/dx #; so #f'(x)=(x^3+2x+1)d/dx(2+x^-2)+d/dx(x^3+2x+1)(2+x^-2)# #f'(x)=(x^3+2x+1)(-2x^-3)+(3x^2+2)(2+x^-2)# #f'(x)=(-2)(x^3+2x+1)/(x^3)+(6x^2+3x^-2+4+2x^-2)# #f'(x)=(-2)(1+2x^-2+x^-3)+(6x^2+4+5x^-2)# #f'(x)=-2-4x^-2-2x^-3+6x^2+4+5x^-2# #f'(x)=6x^2-x^-2-2x^-3-2# 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 1354 views around the world You can reuse this answer Creative Commons License