Question

If `f(x)= csc 3 x` and `g(x) = sqrt(2x-3`, how do you differentiate `f(g(x))` using the chain rule?

Answer 1

`f' (g(x))=-3/sqrt(2x-3)*csc (3sqrt(2x-3) )*cot (3sqrt(2x-3))`

Explanation:

Given `f(x)=csc 3x` and `g(x) = sqrt(2x-3)`

`f(g(x))= csc 3g(x)`
`f(g(x))=csc 3sqrt(2x-3)`

The formula for derivative of `csc u:`

`d/dx(csc u) = - csc u*cot u* (du)/dx`

Let `u=3sqrt(2x-3)`

Take note:
`d/dx(3 sqrt(2x-3))=3* 1/(2sqrt(2x-3))*2`

`f '(g(x))=`
`- csc (3 sqrt(2x-3)) * cot (3 sqrt(2x-3)) * 3* 1/(2sqrt(2x-3))*2`

`f' (g(x))=-3/sqrt(2x-3)*csc (3sqrt(2x-3) )*cot (3sqrt(2x-3))`