Expert: Paul Klarreich - 12/31/2007

Question
y=2sin^3(2x^4)
I need to find to the derivative. I tried spliting it into y = 2sin(2x^4) * sin(2x^4) * sin(2x^4)
but I am not sure that is allowed and I have never taken the derivative of three products. Im officially stuck, can you please help me?

Answer
Questioner:   Catia
Category:  Calculus
Private:  No
 
Subject:  Calculus
Question:  y=2sin^3(2x^4)
I need to find to the derivative. I tried spliting it into y = 2sin(2x^4) * sin(2x^4) * sin(2x^4)
but I am not sure that is allowed and I have never taken the derivative of three products. Im officially stuck, can you please help me?
========================================================
Hi, Catia,

Yes, there is a general product rule:

D(uvw) = u'vw + uv'w + uvw'

but that is not the way to do this.  Try the Chain Rule.  Write:

y = 2u^3,  u = sin(v),  v = 2x^4

dy/du = 6u^2,  du/dv = cos v,  dv/dx = 8x^3

Now

dy   dy du dv
-- = -- -- -- =
dx   du dv dx
 
6u^2 cos v(8x^3)

= 48 x^3 sin^2(2x^4) cos(2x^4)