find the derivative of : (sin(9x))^ln(x)

Question

find the derivative of : (sin(9x))^ln(x)

Answer 1

y=sin(9x)^(ln(x))

 

ln(y)=ln(x)*ln(sin(9x))

Take Derivative now 

(1/y)dy/dx=ln(sin(9x))/x+9*cos(9x)*ln(x)/sin(9x)

       Substitute y       

       dy/dx=[ln(sin(9x))/x+9*cos(9x)*ln(x)/sin(9x)]*sin(9x)^(ln(x))

       simplifying

dy/dx=[ln(sin(9x))/x+9*cot(9x)*ln(x))]*sin(9x)^(ln(x))

Answer 2

Let   y = sin(9x)^ln(x)

 

Ln both sides of the equation.

 

Ln(y) = ln(x)Ln(sin(9x))

 

 

Differentiate both sides of the equation using the chain rule.

 

y' / y = (1/x)Ln(sin(9x)) + ln(x)[9cos(9x) / sin(9x)]

 

y' / y = (1/x)Ln(sin(9x)) + 9ln(x)cot(9x)

 

y' = y[(1/x)Ln(sin(9x)) + 3ln(x)cot(9x)]