If y = tan-1 when 0 < a x < 1, prove that is independent of a.

Question

If y = tan^-1 $open parentheses fraction numerator straight x to the power of begin display style 1 third end style end exponent plus straight a to the power of begin display style 1 third end style end exponent over denominator 1 minus straight x to the power of begin display style 1 third end style end exponent space straight a to the power of begin display style 1 third end style end exponent end fraction close parentheses$ when 0 < a x < 1, prove that $dy over dx$ is independent of a.

Solution

space space space space space straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator straight x to the power of begin display style 1 third end style end exponent plus straight a to the power of begin display style 1 third end style end exponent over denominator 1 minus straight x to the power of begin display style 1 third end style end exponent space straight a to the power of begin display style 1 third end style end exponent end fraction close parentheses space semicolon space Put space straight x to the power of 1 third end exponent equals tan space straight A comma space straight a to the power of 1 third end exponent equals tan space straight B space space space space space straight y equals tan to the power of negative 1 end exponent open square brackets fraction numerator tan space straight A plus tan space straight B over denominator 1 minus tan space straight A space tan space straight B end fraction close square brackets equals tan to the power of negative 1 end exponent left square bracket tan left parenthesis straight A plus straight B right parenthesis right square bracket equals straight A plus straight B therefore space straight y equals tan to the power of negative 1 end exponent straight x to the power of 1 third end exponent plus tan to the power of negative 1 end exponent straight a to the power of 1 third end exponent therefore space dy over dx equals fraction numerator 1 over denominator 1 plus straight x to the power of begin display style 2 over 3 end style end exponent end fraction straight d over dx open parentheses straight x to the power of 1 third end exponent close parentheses plus 0 equals fraction numerator 1 over denominator 1 plus straight x to the power of begin display style 2 over 3 end style end exponent end fraction.1 third straight x to the power of negative begin inline style 2 over 3 end style end exponent comma space which space is space independent space of space straight a.

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