Question

Use logarithmic differentiation to find the derivative of the function y = (cosx)^x

Answer 1

take the log both sides.
lny=xln(cosx)

Answer 2

and take derivative both sides

Answer 3

y prime/y= (ln(cosx))+x(1/cosx)(-sinx) and isolate y prime and you are done

Answer 4

\[\int\limits \text{Cos}[x]^x (\text{Log}[\text{Cos}[x]]-x \text{Tan}[x])dx = \text{Cos}[x]^x \]

Answer 5

put ln on both sides to get \[\ln y =x \ln cosx\] then differentiate to get \[1/y =1\times -sinx\]

Answer 6

also you can use formula for logarithmic differentiation...
d/dx(u^v)=u^v*d/dx(v*log u )