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Continuity And Differentiability

Question

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$Find space dy over dx space where space xy space log space left parenthesis straight x plus straight y right parenthesis equals 1$

Solution

Here space space xy space log space left parenthesis straight x plus straight y right parenthesis equals 1 Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma xy dy over dx left square bracket log left parenthesis straight x plus straight y right parenthesis right square bracket plus straight x space log space left parenthesis straight x plus straight y right parenthesis. straight d over dx left parenthesis straight y right parenthesis plus straight y space log space left parenthesis straight x plus straight y right parenthesis straight d over dx left parenthesis straight x right parenthesis equals 0 therefore xy fraction numerator 1 plus begin display style dy over dx end style over denominator straight x plus straight y end fraction plus straight x space log space left parenthesis straight x plus straight y right parenthesis. dy over dx plus straight y space log space left parenthesis straight x plus straight y right parenthesis.1 equals 0 therefore xy plus xy dy over dx plus straight x left parenthesis straight x plus straight y right parenthesis space log left parenthesis straight x plus straight y right parenthesis dy over dx plus straight y left parenthesis straight x plus straight y right parenthesis log left parenthesis straight x plus straight y right parenthesis equals 0 therefore left square bracket xy plus straight x left parenthesis straight x plus straight y right parenthesis log left parenthesis straight x plus straight y right parenthesis right square bracket dy over dx equals negative left square bracket xy plus straight y left parenthesis straight x plus straight y right parenthesis log left parenthesis straight x plus straight y right parenthesis right square bracket therefore dy over dx equals negative fraction numerator xy plus straight y left parenthesis straight x plus straight y right parenthesis log left parenthesis straight x plus straight y right parenthesis over denominator xy plus straight x left parenthesis straight x plus straight y right parenthesis log left parenthesis straight x plus straight y right parenthesis end fraction

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