can someone help me find the derivative for this function?

Question

y=log(sin^5(x))

Answer 1

y = log(sin⁵(x))

 

Log identity:  Log(a^b) = b*log(a).  Hence:

 

y = log(sin⁵(x)) = 5log(sin(x))

 

Now use the chain rule, letting u = sin(x)

 

y' = dy/dx = 5*dlog(u)/du* du/dx

 

y' = 5*(1/u)*(d(sin(x)/dx)

 

y' = 5*cos(x)/sin(x) = 5cot(x)