Use logarithmic differentiation to find the derivative of the function: `y = (sin 9x)^(lnx)`

Expert Answers
justaguide eNotes educator| Certified Educator

The derivative of `y=(sin 9x)^(ln x)` has to be determined.

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

take the natural logarithm of both the sides

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

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

Take the derivative

`(y')/y = (1/x)*ln(sin 9x) +(ln x*9cos 9x)/(sin 9x)`

=> `y' = y((1/x)*ln(sin 9x) + (ln x*9cos 9x)/(sin 9x))`

=> `y' = (sin 9x)^ln x*(ln(sin 9x)/x + (9cos 9x*ln x)/(sin 9x))`

The derivative `y' = (sin 9x)^(ln x)(ln(sin 9x)/x + (9cos 9x*ln x)/(sin 9x))`