What is the derivative of this function #sin(x) / (1 + sin^2(x))#?

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Answer 1 · 2017-01-19T22:22:43

#cos^3x/(1+sin^2x)^2#

You would use the division rule:

#f'(x)=(cosx(1+sin^2x)-sinx(2sinxcosx))/(1+sin^2x)^2#

#=(cosxcolor(red)(+cosxsin^2x-2sin^2xcosx))/(1+sin^2x)^2##->#sum like terms

#=(color(red)cosx-sin^2xcolor(red)cosx)/(1+sin^2x)^2# #->#factorize

#=(cosx(color(red)(1-sin^2x)))/(1+sin^2x)^2##->#substitute

#=(cosxcolor(red)(cos^2x))/(1+sin^2x)^2##->#multiply

#=cos^3x/(1+sin^2x)^2#