SOLUTION: Use logarithmic differentiation to find the derivative of the function. y=(cos 6x)^x

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Question 804716: Use logarithmic differentiation to find the derivative of the function.
y=(cos 6x)^x
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
y = [cos(6x)]x

Take the natural log of both sides

ln(y) = ln(cos 6x)x

Use the rule ln(AB) = B·ln(A)

ln(y) = x·ln[cos(6x)]

Use the rules 
d%2Fdx[ln(u)] = %22u%27%22%2Fu
d%2Fdxu·v = u·v' + u'·v
d%2Fdxcos(u) = -sin(u)·u'

%22y%27%22%2Fy = x·%28-sin%286x%29%2A6%29%2Fcos%286x%29 + 1·ln[cos(6x)]

%22y%27%22%2Fy = -6x·%28sin%286x%29%29%2Fcos%286x%29 + ln[cos(6x)]

Since sin%28theta%29%2Fcos%28theta%29=tan%28theta%29,

%22y%27%22%2Fy = -6x·tan(6x) + ln[cos(6x)]

Multiply both sides by y

y' = y{-6x·tan(6x) + ln[cos(6x)]}

Go back to the original equation for y, and substitute (cos 6x)x
for y.

y' = [cos(6x)]x{-6x·tan(6x) + ln[cos(6x)]}

Edwin