What is the derivative of h(x)=x(sin(x))^x?

Question

 What is the derivative of h(x)=x(sin(x))^x?

Answer 1

h(x)=x(sin(x))^x

Take logs of both sides.

Ln h(x) = Ln x + x Ln sin x

Take the derivative of both sides.

h' (x). 1/ h(x) = 1/x  + Ln sin x  + x cosx /sin x

h'(x) = h(x) . (1/x + Ln sin x + x cot x)

        = x(sin(x))^x [1/x + Ln sin x + x cot x]

 

Answer 2

let h(x)=y.

 

y = x(sin(x))^x

 

 

Ln both sides of the equation. Bring the x as the coefficient of the ln.

 

ln(y) = lnx(sin(x))^x

 

ln(y) = xlnx(sin(x))

 

 

Now we use implicit differentiation.

 

y' / y = (lnx + x/x)sin(x) + xlnx(cos(x))

 

y' / y = (lnx + 1)sin(x) + xlnx(cos(x))

 

y' / y = lnxsin(x) + sin(x) + xlnxcos(x)

 

 

Finally, substitute the value of y into the derivative.

 

y' = [x(sin(x))^x] * [(lnxsin(x) + sin(x) + xlnxcos(x)]