Question

What is the derivative of `-csc(x)`?

Answer 1

`d/dx (-csc(x)) = -d/dx(csc(x))`
Cosecant is the reciprocal of the sine function, so:
`-d/dx(1/sin(x))`
You should know that the derivative of `1/x = -1/x^2`.
So by the chain rule: `1/f(x) = -1/f(x)^2*d/dxf(x)`

Applying this here:
`-(-1/(sin^2(x))*d/dxsin(x))`
`= -(-1/(sin^2(x))*cos(x))`
`= cos(x)/(sin^2(x))d`
This is already an answer but in most textbooks, you see it in a different form:

`1/sin(x) * cos(x)/sin(x)`
`= csc(x)*cot(x)`

I hope this helped you in any way.