Question

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

Answer 1

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")`