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

1 Answer
Dec 1, 2016

#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.