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Continuity And Differentiability

Question
CBSEENMA12034872

Find space dy over dx comma space if space straight y to the power of straight x plus straight x to the power of straight y plus straight x to the power of straight x equals straight a to the power of straight b.

Solution
We space have space straight y to the power of straight x plus straight x to the power of straight y plus straight x to the power of straight x equals straight a to the power of straight b
Put space straight x to the power of straight y equals straight u comma space straight y to the power of straight x equals straight v
therefore space straight u plus straight v equals straight a to the power of straight b
Differentiating space straight w. straight r. straight t. straight x. space du over dx plus dv over dx equals 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis
Now space straight u equals straight x to the power of straight y space space space space space space space space space space space space space space rightwards double arrow space log space straight u equals straight y space log space straight x
Differentiating space straight w. straight r. straight t. straight x comma space we space get
1 over straight u du over dx equals straight y.1 over straight x plus log space straight x. dy over dx space space rightwards double arrow space du over dx equals straight x to the power of straight y open square brackets straight y over straight x plus log space straight x. dy over dx close square brackets
rightwards double arrow space du over dx equals yx to the power of straight y minus 1 end exponent plus straight x to the power of straight y space log space straight x. dy over dx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis
Also space straight v equals straight y to the power of straight x
therefore space log space straight v equals log space straight y to the power of straight x space space space space space space space space space space space space space space space or space log space straight v equals straight x space log space straight y
Differentiating space straight w. straight r. straight t. straight x.
space space 1 over straight v dv over dx equals straight x space 1 over straight y dy over dx plus log space straight y.1
therefore space dv over dx equals straight y to the power of straight x
therefore space dv over dx equals straight x space straight y to the power of straight x minus 1 end exponent dy over dx plus straight y to the power of straight x. log space straight y space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis
From space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis. space we space get.
space space space space space space space yx to the power of straight y minus 1 end exponent plus straight x to the power of straight y log space straight x. dy over dx plus xy to the power of straight x minus 1 end exponent dy over dx plus straight y to the power of straight x. log space straight y equals 0
therefore space left parenthesis straight x to the power of straight y log space straight x plus xy to the power of straight x minus 1 end exponent right parenthesis dy over dx equals negative left parenthesis yx to the power of straight y minus 1 end exponent plus straight y to the power of straight x. log space straight y right parenthesis
therefore space dy over dx equals negative open parentheses fraction numerator yx to the power of straight y minus 1 end exponent plus straight y to the power of straight x. log space straight y over denominator straight x to the power of straight y log space straight x plus xy to the power of straight x minus 1 end exponent end fraction close parentheses.

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