Expert: Paul Klarreich - 10/29/2005

Question
find the derivative of y with respect to the given independent variable.

1) y=log7((sinxcosx)/(e^x2^x))

Answer
Hi, Amy,

Rules:

log(base a) x = ln(x)/ln(a)

So log7(...) = ln(...)/ln 7

Also  since a = exp(ln a)  you can always write  
a^x = exp(x ln a)

So write exp(x) 2^x = exp(x) exp(x ln 2) = exp(x(1 + ln 2))
And  ln(exp(x(1 + ln 2)) = x(1 + ln 2)

So in your example:

y = log7((sinxcosx)/(e^x2^x))

  ln[(sin x cos x)/(exp(x) exp(x ln 2))]
= ---------------------------------------
              ln 7
 ln(sin x) + ln(cos x) - x(1 + ln 2)
= -----------------------------------
             ln 7

From here you should be OK.  And note that this is NOT a quotient rule example, because the denominator is a constant.