Question

$fraction numerator straight d squared straight x over denominator dy squared end fraction$ equal to
$open parentheses fraction numerator straight d squared straight x over denominator dy squared end fraction close parentheses to the power of negative 1 end exponent$
$negative open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses to the power of negative 1 end exponent open parentheses dy over dx close parentheses to the power of negative 3 end exponent$
$open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses open parentheses dy over dx close parentheses to the power of negative 2 end exponent$
$negative open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses open parentheses dy over dx close parentheses to the power of negative 3 end exponent$

Solution

negative open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses open parentheses dy over dx close parentheses to the power of negative 3 end exponent

dy over dx space equals space fraction numerator 1 over denominator begin display style dx over dy end style end fraction fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space straight d over dx open parentheses fraction numerator 1 over denominator dx divided by dy end fraction close parentheses space equals space fraction numerator begin display style straight d end style over denominator begin display style dx end style end fraction open parentheses fraction numerator begin display style 1 end style over denominator begin display style dx divided by dy end style end fraction close parentheses. fraction numerator begin display style straight d end style over denominator begin display style dx end style end fraction equals space minus space space fraction numerator begin display style 1 end style over denominator begin display style open parentheses dx over dy close parentheses squared end style end fraction. fraction numerator begin display style fraction numerator straight d squared straight x over denominator dy squared end fraction end style over denominator begin display style dx over dy end style end fraction space equals space fraction numerator begin display style negative fraction numerator straight d squared straight x over denominator dy squared end fraction end style over denominator begin display style open parentheses dx over dy close parentheses cubed end style end fraction equals negative open parentheses fraction numerator begin display style straight d squared straight x end style over denominator begin display style dy squared end style end fraction close parentheses open parentheses fraction numerator begin display style dy end style over denominator begin display style dx end style end fraction close parentheses cubed

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