Expert: Ahmed Salami - 8/24/2010

Question
Hi,I have a problem with differentiating this function:
u(x,y) =(5x^2 − 8x − 6)^sin(6y)
Find: Ux and Uy.
I thought of using logarithmic method. But don't know how to go about it.
Your help is very much appreciated =)


Answer
Hi Caleb,
u = (5x² - 8x - 6)^sin6y
Let p = 5x² - 8x - 6
u = p^sin6y
Up = sin6y . p^(sin6y - 1)
dp/dx = 10x - 8
Ux = Up . dp/dx
     = [sin6y . (5x² - 8x - 6)^(sin6y - 1)] . (10x - 8)

Let q = sin6y
u = (5x² - 8x - 6)^q
Taking natural logarithm of both sides,
ln u = ln (5x² - 8x - 6)^q
       = q . ln (5x² - 8x - 6)
(1/u) . Uq = ln (5x² - 8x - 6)
Uq = u . ln (5x² - 8x - 6)
     = ln (5x² - 8x - 6) . (5x² - 8x - 6)^sin6y
dq/dy = 6 . cos6y
Uy = Uq . dq/dy
     = [ln (5x² - 8x - 6) . (5x² - 8x - 6)^sin6y] . (6cos6y)

You can always get back to me.

Regards