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

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

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