Question

Question #bd1e6

Answer 1

The quotient rule states that

`d/dx f(x)/g(x) = (f'(x)g(x)-f(x)g'(x))/[g(x)]^2`

As `cot(x) = cos(x)/sin(x)`, we can apply the quotient rule with `f(x)=cos(x)` and `g(x)=sin(x)` to get

`d/dxcot(x) = d/dxcos(x)/sin(x)`

`=(sin(x)(d/dxcos(x))-cos(x)(d/dxsin(x)))/sin^2(x)`

`=(sin(x)(-sin(x))-cos(x)(cos(x)))/sin^2(x)`

`=(-sin^2(x)-cos^2(x))/sin^2(x)`

`=-(sin^2(x)+cos^2(x))/sin^2(x)`

`=-1/sin^2(x)`

`=-csc^2(x)`