Question

If `f(x) =sin^3x` and `g(x) = sqrt(3x-1`, what is `f'(g(x))`?

Answer 1

`f(x)=sin^3x` , `D_f=RR`

`g(x)=sqrt(3x-1)`, `Dg=[1/3,+oo)`

`D_(fog)={` `AAx` `in` `RR:` `x` `in` `D_g`, `g(x)` `in` `D_f}`

`x>=1/3` , `sqrt(3x-1)` `in` `RR` `->` `x` `in` `[1/3,+oo)`

`AAx` `in` `[1/3,+oo)`,

• `(fog)'(x)=f'(g(x))g'(x)=f'(sqrt(3x-1))((3x-1)')/(2sqrt(3x-1))`

`f'(x)=3sin^2x(sinx)'=3sin^2xcosx`

so `(fog)'(x)=sin^2(sqrt(3x-1))cos(sqrt(3x-1))*9/(2sqrt(3x-1))`