Question

$If space straight y space equals space log space open square brackets straight x space plus space square root of straight x squared plus space straight a squared end root close square brackets comma space show space that space left parenthesis straight x squared space plus space straight a squared right parenthesis space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus straight x dy over dx space equals space 0$

Solution

Given,

$straight y space equals space log space open square brackets straight x space plus space square root of straight x squared space plus space straight a squared end root close square brackets space straight y space equals space log space open square brackets straight x space plus space square root of straight x squared space plus straight a squared end root close square brackets...... space left parenthesis straight i right parenthesis Differentiating space left parenthesis 1 right parenthesis space straight w. straight r. straight t space space straight x comma space we space get dy over dx space equals space fraction numerator 1 plus fraction numerator straight x over denominator square root of straight x squared plus straight a squared end root end fraction over denominator straight x space plus space square root of straight x squared space plus space straight a squared end root end fraction rightwards double arrow space dy over dx space equals space fraction numerator 1 over denominator square root of straight x squared space plus space straight a squared end root end fraction space space space space space..... space left parenthesis ii right parenthesis rightwards double arrow space straight x dy over dx space equals space fraction numerator straight x over denominator square root of straight x squared space plus space straight a squared end root end fraction space space space space space space.... space left parenthesis iii right parenthesis Again comma space differentiating space left parenthesis ii right parenthesis space straight w. straight r. straight t space straight x comma space we space get fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space fraction numerator negative begin display style fraction numerator 2 straight x over denominator 2 left parenthesis straight x squared space plus space straight a squared right parenthesis to the power of begin display style 1 half end style end exponent end fraction end style over denominator left parenthesis straight x squared space plus space straight a squared right parenthesis end fraction rightwards double arrow space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space minus fraction numerator straight x over denominator left parenthesis straight x squared space plus space straight a squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction rightwards double arrow space left parenthesis straight x squared space plus space straight a squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space minus fraction numerator straight x over denominator square root of straight x squared space plus space straight a squared end root end fraction space space... space left parenthesis iv right parenthesis adding space equation space left parenthesis iii right parenthesis space and space left parenthesis iv right parenthesis comma space we space get left parenthesis straight x squared space plus space straight a squared right parenthesis space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus space straight x dy over dx space equals space minus fraction numerator straight x over denominator square root of straight x squared plus straight a squared end root end fraction space space plus fraction numerator straight x over denominator square root of straight x squared space plus space straight a squared end root end fraction space equals space 0 rightwards double arrow space left parenthesis space straight x squared space plus space straight a squared right parenthesis space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus space straight x dy over dx space equals space 0$

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