Question

What is the derivative of `(sin x)^(3x)`?

Answer 1

`3(sinx)^(3x)(lnsinx+xcotx)`

Explanation:

`y=(sinx)^(3x)`
taking ln both sides
`lny=ln(sinx)^(3x)`
`lny=(3x)ln(sinx)`
Differentiate both sides
`(1/y)dy/dx=3lnsinx+3x(1/sinx)cosx`
`dy/dx=(3lnsinx+3xcotx)y`
putting value of y
`dy/dx=3(lnsinx+xcotx)(sinx)^(3x)`
it can be written as
`dy/dx=3(sinx)^(3x)(lnsinx+xcotx)`