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Continuity And Differentiability

Question

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Differentiate the following functions w.r.t.x:
$straight x cubed plus straight y cubed equals sin left parenthesis straight x plus straight y right parenthesis$

Solution

Here space space space space space space space space straight x cubed plus straight y cubed equals sin left parenthesis straight x plus straight y right parenthesis Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma space space space space space 3 straight x squared plus 3 straight y squared dy over dx equals cos left parenthesis straight x plus straight y right parenthesis. open parentheses 1 plus dy over dx close parentheses therefore space 3 straight x squared plus 3 straight y squared dy over dx equals cos left parenthesis straight x plus straight y right parenthesis plus cos left parenthesis straight x plus straight y right parenthesis dy over dx therefore space left square bracket 3 straight y squared minus cos left parenthesis straight x plus straight y right parenthesis right square bracket dy over dx equals cos left parenthesis straight x plus straight y right parenthesis minus 3 straight x squared therefore space space space space dy over dx equals fraction numerator cos left parenthesis straight x plus straight y right parenthesis minus 3 straight x squared over denominator 3 straight y squared minus cos left parenthesis straight x plus straight y right parenthesis end fraction

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