Question

$Given space that space straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator 1 minus 2 straight x end fraction close parentheses comma space minus 1 less than straight x less than 1 half. space After space using space the space property space of space inverse space trigonometric space function comma space show space that space dy over dx equals 0$

Solution

Here space straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses plus tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator 1 minus 2 straight x end fraction close parentheses equals tan to the power of negative 1 end exponent 1 minus tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent 2 equals tan to the power of negative 1 end exponent 1 plus tan to the power of negative 1 end exponent 2 therefore space dy over dx equals 0 plus 0 space rightwards double arrow space dy over dx equals 0

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