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What is the derivative of tan^-1(cot(sqrt(x)))?

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What is the derivative of tan^-1(cot(sqrt(x)))?

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We have the function f(x) = arc tan ( cot ( sqrt x)) and we have to find the derivative of f(x).

arc tan ( cot ( sqrt x))

we know that cot x = tan ( pi/2 - x)

=> arc tan ( tan ( pi/2 - sqrt x))

=> pi/2 - sqrt x

f(x) = pi/2 - sqrt x

=> f(x) = pi/2 - x^(1/2)

f'(x) = 0 - (1/2)* x^(1/2 - 1)

=> -(1/2) x^(-1/2)

=> (-1/2)/x^(1/2)

=> -1 / 2*sqrt x

The required derivative is -1/(2*sqrt x)

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