Question

How do you find the derivative of `f(x)=csc(3x-1)`?

Answer 1

`f'(x)=3cos(3x-1)csc^2(3x-1)`

Explanation:

`f(x)` is a xomposite of two functions
Let :
`color(blue)(g(x)=3x-1) and color(brown)(h(x)=cscx`
So,
`f(x)=color(brown)hcolor(blue)((g(x))`
so thederivative of this function is by applying chain rule:

`color(red)(f'(x))=color(red)(h'(g(x)*(g'(x))`

`color(red)h'(g(x))=????`

`color(brown)(h(x)=cscx`
`h'(x)=cosx*csc^2x`
`color(red)h'(g(x))=cos(g(x))*csc^2(g(x))=cos(3x-1)*csc^2(3x-1)`

`color(red)(h'(g(x))) = cos(3x-1)*csc^2(3x-1)`

`color(red)(g'(x))=????`

`color(blue)(g(x)=3x-1)`
`color(red)(g'(x))=3`

`color(red)(f'(x))=color(red)(h'(g(x)*(g'(x))`
`color(red)(f'(x))=cos(3x-1)*csc^2(3x-1)*3`

`color(red)(f'(x)=3cos(3x-1)csc^2(3x-1))`