Expert: Paul Klarreich - 12/22/2006

Question
Using the product rule to differentiate:

f(x)=(cosx)ln^(xsqrt3)

Answer
Questioner:  Duncan
Category:  Calculus
 
Subject:  Product Rule to Diff an Equation
Question:  Using the product rule to differentiate:

f(x)=(cosx)ln^(xsqrt3)
..............................
Hi, Duncan,

As usual in these examples, you will need more than the product rule.  I suggest you use it this way:

Let  f(x) = uv, where

u = cos x
v = ln(x sqrt(3))

[I assume the '^' in the example was a misclick, because  ln^(something) does not have any logical meaning.]

Now differentiate each part separately, so you don't get mixed up trying to do many things at once:

du/dx = - sin x

v = ln (x sqrt(3)) = ln x + ln (sqrt(3))

dv/dx = 1/x  [remember, ln(sqrt(3)) is a constant.]

Ready to go now:

f'(x) = u dv/dx + v du/dx

= (cos x)(1/x) + (- sin x)( ln(x sqrt(3)) )
  cos x
= -------  - sin x ln(x sqrt(3)) )
    x

and that's about it.