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y=(x+cos x)/(x+sin x) , find dy/dx...

`y=(x+cos x)/(x+sin x)` , find `dy/dx `

Text Solution

Verified by Experts

The correct Answer is:
`(cos x - sinx + x(cos x + sin x)+1)/((x+ cos x)^(2))`
Using quotient rule, we have
`(dy)/(dx)=(d)/(dx)((x+ sin x)/(x+ cos x))`
`=((x+ cos x).(d)/(dx)(x+ sin x)-(x + sin x).(d)/(dx)(x+ cos x))/((x+ cos x)^(2))`
`=((x+cos x)cdot(1 + cos x)-( x+ sin x)cdot(1-sin x))/((x+cos x)^(2))`
`=(x+cos x + x cos x +cos^(2)x-x- sin x + x sin x + sin^(2)x)/((x+ cos x )^(2))`
`=(cos x - sin x + x cos x + x sin x + cos^(2) x + sin ^(2) x)/((x+ cos x)^(2))`
`=(cos x - sin x + x (cos x + sin x)+1)/((x+ cos x)^(2))`
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