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

1 Answer
Dec 3, 2016

#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!