How do you differentiate #f(x)= 3x^2 (4x - 12)^2# using the product rule?

1 Answer
Nov 12, 2015

Answer:

Taken you to a point where you should be able to complete the calculation.

Explanation:

Let #u =3x^2" "# then #(du)/(dx) =6x# ..............(1)

Let #v =(4x-12)^2" "# then #(dv)/(dx) =8(4x-12) =32x -96#.....(2)

Product #(v (du)/(dx) + u (dv)/(dx))" "#

#color(blue)("~~~~~~~~~Foot Note showing How to obtain (2) ~~~~~~~~~~~~~~")#

#color(brown)("Let "w=4x-12)#

#color(brown)((dw)/(dx)=4)#

#color(brown)("But "v=w^2 " so " (dv)/(dw) = 2w)#

#color(brown)("But "(dv)/(dx) = (dv)/(dw) times (dw)/(dx)) #

#color(brown)("(dv)/(dx)= 2w times 4 = 8w = 8(4x-12))#

#color(blue)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")#

By substitution we have:

#{(4x-12)^2 times 6x} + {3x^2 times (32x-96)}#

#color(blue)("I have left the final calculation and simplification for you to complete")#