50 POINTS !!!!! HomeWorker Helperer please 1) Using the quotient rule, find y = (cosx - sinx)/(cosx + sinx) 2) Given that y = (x-1)/(x+1) , prove that (x+1)((d^(2)y )/(dx^(2) ))…

Question

50 POINTS !!!!! HomeWorker Helperer please

1) Using the quotient rule, find
`y = (cosx - sinx)/(cosx + sinx)`

2) Given that
`y = (x-1)/(x+1)` , prove that
`(x+1)((d^(2)y )/(dx^(2) )) + 2((dy)/(dx)) = 0`

Answer 1

1) - 2 / (cos x + sinx)^2.

Explanation:

1) I assume you want the derivative of (cos x - sin x) / ( cos x + sin x).

y' = (cosx + sinx)( -sinx - cosx) - (cosx - sinx)(-sinx + cosx)

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(cosx + sinx)^2

= -sinxcosx - cos^2x - sin^2x - sinxcosx - (-sinxcosx+cos^2x+sin^2x-sinxcosx

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(cosx + sinx)^2

= -2sinxcosx - 1 - ( -2sinxcosx + 1 ) / (cosx + sinx)^2

= -2sinxcosx - 1 - ( -2sinxcosx + 1 ) / (cosx + sinx)^2

= - 2 / (cos x + sinx)^2.