Question

How do you differentiate `sin(x) / (1 + sin^2(x))`?

Answer 1

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

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`