Question

How do you differentiate `f(x)=cosx/sinx`?

Answer 1

`f'(x)=-csc^2x`

Explanation:

differentiate f(x) using the `color(blue)"quotient rule"`

`color(orange)"Reminder"`

`"If " f(x)=(g(x))/(h(x)) " then"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2)color(white)(2/2)|)))`

`g(x)=cosxrArrg'(x)=-sinx`

`h(x)=sinxrArrh'(x)=cosx`

`rArrf'(x)=(sinx(-sinx)-cosx(cosx))/sin^2x`

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

`=-1/sin^2x=-csc^2x`

`color(blue)"NOTE"`

`cosx/sinx=cotx`

`" and " d/dx(cotx)=-csc^2x` is a standard result, well worth remembering.