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calyne
Use logarithmic differentiation to find the derivative of the function y = (cosx)^x
take the log both sides. lny=xln(cosx)
and take derivative both sides
y prime/y= (ln(cosx))+x(1/cosx)(-sinx) and isolate y prime and you are done
\[\int\limits \text{Cos}[x]^x (\text{Log}[\text{Cos}[x]]-x \text{Tan}[x])dx = \text{Cos}[x]^x \]
put ln on both sides to get \[\ln y =x \ln cosx\] then differentiate to get \[1/y =1\times -sinx\]
also you can use formula for logarithmic differentiation... d/dx(u^v)=u^v*d/dx(v*log u )