Question

How do you use the chain rule to differentiate `y=-2csc^6x`?

Answer 1

`dy/dx=dy/(dg)(dg)/dx`, where `g(x)` is `cscx`

Explanation:

Make the substitution `g(x)=cscx`
`y=-2g^6(x)`

Now use the chain rule
`dy/dx=dy/(dg)(dg)/dx`
`dy/(dg)=-12g^5(x)`
`(dg)/dx=(cscx)'=-cscxcotx`
Plug these values in the initial equation
`dy/dx=dy/(dg)(dg)/dx=-12g^5(x)g'(x)`
`dy/dx=12csc^5xcscxcotx`
Therefore, the solution is
`dy/dx=12csc^7xcosx`