How do you differentiate #f(x)=(2x^3)(cos^2x^2) # using the product rule? Calculus Basic Differentiation Rules Product Rule 1 Answer IZUkuABA Jun 26, 2018 #2x^3*2*cosx^2*-sin(x)^2*2x+cos^2x^2*6x^2# Explanation: we have #f(x)=(2x^3)(cos^2x^2)# #dy/dx=2x^3*d(cos^2x^2)/dx+(cos^2x^2)*d(2x^3)/dx# #=>2x^3*2*d(cosx^2)/dx+(cos^2x^2)*2*3*x^2#.....using#[dy/dx=x^n=>nx^(n-1)]# #=>2x^3*2*(cosx^2)*-sind(x^2)/dx+(cos^2x^2)*6*x^2# .........using #[dy/dx=cosx=>-sinx]# #=>2x^3*2*(cosx^2)*-sinx^2*2x+(cos^2x^2)*6*x^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 1559 views around the world You can reuse this answer Creative Commons License