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Continuity And Differentiability

Question

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Find $dy over dx$ in the following :
x³ + x² y + x y² + y³ = 81

Solution

Here space straight x cubed space plus space straight x to the power of 2 space end exponent straight y space plus space straight x space straight y squared space plus space straight y to the power of 3 space end exponent equals space 81 Differntiating space both space sides space straight w. straight r. straight t space straight x comma space we space get comma space space space space space space space space straight d over dx left parenthesis straight x cubed right parenthesis plus straight d over dx left parenthesis straight x squared straight y right parenthesis plus straight d over dx left parenthesis xy squared right parenthesis plus straight d over dx left parenthesis straight y cubed right parenthesis equals straight d over dx left parenthesis 81 right parenthesis therefore space space space space 3 straight x squared plus straight x squared dy over dx plus straight y.2. straight x plus straight x.2 straight y dy over dx plus straight y squared.1 plus 3 straight y squared dy over dx equals 0 therefore space space space space left parenthesis straight x squared plus 2 xy plus 3 straight y squared right parenthesis dy over dx equals negative left parenthesis 3 straight x squared plus 2 xy plus straight y squared right parenthesis therefore space space space space space space space space space space space space space dy over dx equals negative open parentheses fraction numerator 3 straight x squared plus 2 xy plus straight y squared over denominator straight x squared plus 2 xy plus 3 straight y squared end fraction close parentheses.

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