How do you differentiate sin(x) / (1 + sin^2(x))?
1 Answer
May 5, 2016
Explanation:
Differentiate using the
color(blue)" quotient rule " If f(x)
=(g(x))/(h(x))"then" f'(x)=(h(x).g'(x)-g(x).h'(x))/(h(x)^2
"--------------------------------------------------------------------" g(x)
=sinxrArrg'(x)=cosx h(x)
=1+sin^2xrArrh'(x)=2sinxcosx
"-------------------------------------------------------------------"
now substitute these values into f'(x)
f'(x)=((1+sin^2x)cosx-sinx(2sinxcosx))/(1+sin^2x)^2
=(cosx+cosxsin^2x-2cosxsin^2x)/(1+sin^2x)^2
=(cosx-cosxsin^2x)/(1+sin^2x)^2=(cosx(1-sin^2x))/(1+sin^2x)^2 and using trig identity
color(red)(|bar(ul(color(white)(a/a)color(black)(sin^2x+cos^2x=1)color(white)(a/a)|))
rArr1-sin^2x=cos^2x
rArrf'(x)=(cos^3x)/(1+sin^2x)^2
