Question

Have a function f defined by


`f(x)=(sin(x)+1)^x`
determine the value of

`f'\left((3\pi)/(2)\right)`

​

Answer 1

`f'((3\pi)/(2))` is undefined

Explanation:

`f(x)=(sinx+1)^x`

`(d)/(dx)(sinx+1)^x`

`(d)/(dx)e^(ln((sinx+1)^x))`

`(d)/(dx)e^(xln(sinx+1))`

`((d)/(dx)(x)*ln(sinx+1)+x*(d)/(dx)ln(sinx+1))e^(xln(sinx+1))`

`(ln(sinx+1)+x*cosx((1)/(sinx+1)))(sinx+1)^x`

`[ln(sinx+1)+(xcosx)/(sinx+1)](sinx+1)^x`

`f'((3\pi)/(2))=[ln(sin(3\pi)/(2) +1)+((3\pi)/(2) cos(3\pi)/(2) )/(sin(3\pi)/(2) +1)](sin(3\pi)/(2) +1)^{(3\pi)/(2)}`

`f'((3\pi)/(2))=[ln(-1+1)+((3\pi)/(2) (0) )/(-1+1)](-1+1)^(-1)`

`f'((3\pi)/(2))=[ln(0)+(0)/(0)](0)^(-1)`

Because the derivative is undefined, then the function isn't differentiable at the point
`((3\pi)/(2),0)`, making it a critical point since the slope of the function is 0.