Expert: Paul Klarreich - 10/29/2005

Question
find the derivative of y with respect to x, t, or theta, as appropriate.

1) y=ln(2e^-tsint)
2) y=ln[(sqrt(theta))/(1+sqrt(theta)]

Answer
Hi, Amy,

FIRST, a request for future problem submissions.  If you have several problems that you can't do and want to send them, PUT THEM ALL INTO ONE QUESTION.  When you make several question submissions on the same topic, you are interfering with the process.  The web site puts a limit on the number of questions that I can answer at a time, so if you send a bunch of questions you fill up the quota and nobody else can submit a question for me.  

As in the last one, remember certain rules for exponential and logarithmic functions, including:

ln(ab) = ln(a) + ln(b)
ln(a/b) = ln(a) - ln(b)
ln(a^n) = n ln(a)

1) y = ln(2 exp(-t sin t)) = ln 2 + ln(exp(-t sin t))
    = ln 2 - t sin t
Now just use the quotient rule on the second term.  (ln 2 is a constant, of course.)

2. y = ln[sqrt(t)/(1 + sqrt(t))] =
   ln(sqrt(t)) - ln(1 + sqrt(t))=
   1/2 ln(t) - ln(1 + sqrt(t))
 [LN RULE]   [LN AND CHAIN RULES]

dy/dt = 1/2t - 1/(1 + sqrt(t))(1/(2sqrt(t)))

dy    1         1          1
-- = --- - ----------- ---------
dy   2t    1 + sqrt(t) 2 sqrt(t)

etc.