Question

What is the derivative of `y=cot^{-1}(x)`?

Answer 1

The answer is `y'=-1/(1+x^2)`

We start by using implicit differentiation:

`y=cot^(-1)x`
`cot y=x`
`-csc^2y (dy)/(dx)=1`
`(dy)/(dx)=-1/(csc^2y)`
`(dy)/(dx)=-1/(1+cot^2y)` using trig identity: `1+cot^2 theta=csc^2 theta`
`(dy)/(dx)=-1/(1+x^2)` using line 2: `cot y = x`

The trick for this derivative is to use an identity that allows you to substitute `x` back in for `y` because you don't want leave the derivative as an implicit function; substituting `x` back in will make the derivative an explicit function.