What is the derivative dy/dx for y=(sin x + cos x)/(sin x - cos x)?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find the derivative of y = (sin x + cos x)/(sin x - cos x)

Use the quotient rule that states: if y = f(x)/g(x), y' = [f'(x)g(x) - f(x)g'(x)]/(g(x))^2

y = (sin x + cos x)/(sin x - cos x)

y' = [(cos x - sin x)(sin x - cos x) - (sin x + cos x)(cos x + sin x)]/(sin x - cos x)^2

=> [cos x*sin x - (cos x)^2 - (sin x)^2 + sin x*cos x -sin x*cos x - (cos x)^2 - (sin x)^2 - sin x*cos x]/(sin x - cos x)^2

=> [-2(cos x)^2 - 2(sin x)^2]/(sin x - cos x)^2

The required derivative is [-2(cos x)^2 -2(sin x)^2]/(sin x - cos x)^2

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