Using logarithmic differentiation to solve derivative

Question

Hi, I need to solve this by log differentiation: 

 

[ sin(x) ]^x² 

Answer 1

Let    y = [sin(x)]^x²

 

Ln both sides of the equation.  Bring all exponents as the coefficient of the log.

 

Ln(y) = x²Ln(sin(x))

 

 

Differentiate both sides of the equation using the chain rule.

 

y' / y = 2xLn(sin(x)) + x²(cos(x)/sin(x))

 

y' / y = 2xLn(sin(x)) + x²cot(x)

 

 

Solving for y'

 

y' = y[2xLn(sin(x)) + x²cot(x)]

 

 

Finally, substitute the original value of y into this derivative.

 

y' = sin(x)^x²[2xLn(sin(x)) + x²cot(x)]