Question

What is the derivative of this function `-1/(sin(x))^2`?

Answer 1

`dy/dx = 2cotxcsc^2x`

Explanation:

This function can be written as `y = -(sinx)^-2`.

We let `y = -u^-2` and `u= sinx`. Through the power rule and basic differentiation rules, we obtain `dy/(du)= -2(-1)u^-3 = 2/u^3` and `(du)/dx = -cosx`.

Recall that the chain rule for derivatives of composite functions states that `color(red)(dy/dx = dy/(du) xx (du)/dx)`.

`dy/dx = 2/u^3 xx cosx`

`dy/dx = (2cosx)/(sinx)^3`

By the identities `cotx = 1/tanx` and `cscx = 1/sinx`, we can simplify the derivative.

`dy/dx = 2cotxcsc^2x`

Hopefully this helps!