Question

$If space straight theta equals cos to the power of negative 1 end exponent open parentheses straight r over straight k close parentheses minus fraction numerator square root of straight k squared minus straight r squared end root over denominator straight r end fraction comma space find space dθ over dr$

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Solution

Here space straight theta equals cos to the power of negative 1 end exponent open parentheses straight r over straight k close parentheses minus fraction numerator square root of straight k squared minus straight r squared end root over denominator straight r end fraction Differentiating space both space sides space straight w. straight r. straight t. space straight theta. space we space get comma dθ over dr equals negative fraction numerator 1 over denominator square root of 1 minus open parentheses begin display style straight r over straight k end style close parentheses squared end root end fraction.1 over straight k minus fraction numerator straight r. begin display style fraction numerator negative 2 straight r over denominator 2 square root of straight k squared minus straight r squared end root end fraction end style minus square root of straight k squared minus straight r squared end root.1 over denominator straight r squared end fraction space space space space space space space space equals negative fraction numerator straight k over denominator square root of straight k squared minus straight r squared end root end fraction.1 over straight k plus fraction numerator begin display style fraction numerator straight r squared over denominator square root of straight k squared minus straight r squared end root end fraction end style plus begin display style fraction numerator square root of straight k squared minus straight r squared end root over denominator 1 end fraction end style over denominator straight r squared end fraction space space space space space space space space equals negative fraction numerator 1 over denominator square root of straight k squared minus straight r squared end root end fraction plus fraction numerator straight r squared plus straight k squared minus straight r squared over denominator straight r squared square root of straight k squared minus straight r squared end root end fraction equals negative fraction numerator 1 over denominator square root of straight k squared minus straight r squared end root end fraction plus fraction numerator straight k squared over denominator straight r squared square root of straight k squared minus straight r squared end root end fraction space space space space space space space space equals fraction numerator negative straight r squared plus straight k squared over denominator straight r squared square root of straight k squared minus straight r squared end root end fraction equals fraction numerator square root of straight k squared minus straight r squared end root over denominator straight r squared end fraction

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