Differentiate the following: `y = cos^(2) (1 - sqrt(x))/(1 + sqrt(x))` Answer should be:  `1/ (sqrt(x)(1 + sqrt(x))^(2)) sin (2(1 - sqrt x)/(1+ sqrt(x)))`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The expression `y = cos^2((1-sqrtx)/(1+sqrtx))` has to be differentiated.

y = `cos^2((1 - sqrtx)/(1+sqrtx))`

= `(1+cos(2*(1 - sqrtx)/(1+sqrtx)))/2`

y' = `(1/2)*-sin(2*(1 - sqrtx)/(1+sqrtx))*2*((1 - sqrtx)/(1+sqrtx))'`

= `-sin(2*(1 - sqrtx)/(1+sqrtx))*((1 - sqrtx)/(1+sqrtx))'`

= `-sin(2*(1 - sqrtx)/(1+sqrtx))*((1 - sqrtx)'(1+sqrt x) - (1-sqrt x)*(1+sqrt x)')/(1+sqrtx)^2`

= `-sin(2*(1 - sqrtx)/(1+sqrtx))*((-1/2)*(1/sqrt x)(1+sqrt x) - (1-sqrt x)*(1/2)*(1/sqrtx))/(1+sqrt x)^2`

= `-sin(2*(1 - sqrtx)/(1+sqrt x))*(-1/2)*(1/sqrtx)((1+sqrtx + 1-sqrt x))/(1+sqrt x)^2`

= `sin(2*(1 - sqrtx)/(1+sqrt x))*(1/2)*(1/sqrtx)(2/(1+sqrt x)^2)`

= `sin(2*(1 - sqrtx)/(1+sqrt x))*(1/sqrtx)(1/(1+sqrt x)^2)`

= `sin(2*(1 - sqrtx)/(1+sqrt x))*1/(sqrtx*(1+sqrt x)^2)`

The derivative of `y = cos^2((1-sqrtx)/(1+sqrt x))` is `y' = sin(2*(1 - sqrtx)/(1+sqrt x))*1/(sqrtx*(1+sqrt x)^2)`

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