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Continuity And Differentiability

Question
CBSEENMA12035635
Wired Faculty App

if space cos space left parenthesis straight a plus straight y right parenthesis space equals space cos space straight y space then space prove space that space dy over dx space equals fraction numerator cos squared left parenthesis straight a plus straight y right parenthesis over denominator sin space straight a end fraction Hence space show space that space sin space straight a space fraction numerator straight d squared straight y over denominator dx squared end fraction plus sin space 2 space left parenthesis straight a plus straight y right parenthesis dy over dx equals 0

Solution
Given space that comma straight x space cos space left parenthesis straight a plus straight y right parenthesis space equals space cos space straight y space.... space left parenthesis straight i right parenthesis rightwards double arrow space straight x space equals space fraction numerator cos space straight y over denominator cos space left parenthesis straight a space plus space straight y right parenthesis end fraction space... space left parenthesis ii right parenthesis Differentiating space both space sides space of space the space equation space left parenthesis straight i right parenthesis comma space we space have comma straight x space left parenthesis negative sin space left parenthesis straight a space plus space straight y right parenthesis right parenthesis space dy over dx equals space minus space cos space left parenthesis straight a plus straight y right parenthesis space equals negative sin space straight y space dy over dx rightwards double arrow space space open square brackets sin space straight y minus space straight x space sin space left parenthesis straight a space plus space straight y right parenthesis close square brackets dy over dx space equals space minus cos space left parenthesis straight a space plus space straight y right parenthesis rightwards double arrow space open square brackets sin space straight y space minus fraction numerator cos space straight y over denominator cos space left parenthesis straight a plus straight y right parenthesis end fraction sin space left parenthesis straight a space plus space straight y right parenthesis close square brackets dy over dx space space equals negative cos space left parenthesis straight a space plus space straight y right parenthesis rightwards double arrow space open square brackets fraction numerator cos space left parenthesis straight a space plus space straight y right parenthesis space space straight x space space sin space straight y space minus space cos space straight y space sin space left parenthesis straight a space plus space straight y right parenthesis over denominator cos space left parenthesis straight a plus straight y right parenthesis end fraction close square brackets dy over dx space equals space minus space cos left parenthesis straight a space plus space straight y right parenthesis space rightwards double arrow space open square brackets cos space left parenthesis straight a space plus space straight y right parenthesis space straight x space sin space straight y space minus space cosy space space straight x space space sin space left parenthesis straight a space plus space straight y right parenthesis close square brackets dy over dx equals negative cos left parenthesis straight a space plus space straight y right parenthesis space space straight x space space cos space left parenthesis straight a space plus space straight y right parenthesis rightwards double arrow space space open square brackets sin space left parenthesis straight a space plus space straight y space minus space straight y right parenthesis close square brackets dy over dx space space equals space minus cos squared left parenthesis straight a space plus space straight y right parenthesis space space space space space open square brackets sin left parenthesis straight A minus straight B right parenthesis space equals space sinA space cos space straight B space minus space cosA space sinB close square brackets rightwards double arrow left square bracket sina space right square bracket space dy over dx equals negative cos squared left parenthesis straight a space plus space straight y right parenthesis rightwards double arrow space dy over dx space equals fraction numerator negative cos squared left parenthesis straight a space plus space straight y right parenthesis over denominator sin space straight a space end fraction space space space space.. space left parenthesis iii right parenthesis differentiating space once space again space with space respect space to space straight x space comma we space have comma sin space straight a space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus space 2 cos left parenthesis straight a space plus space straight y right parenthesis space sin space left parenthesis straight a space plus space straight y right parenthesis dy over dx rightwards double arrow space sin space straight a space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus space 2 space cos space left parenthesis straight a space plus space straight y right parenthesis space sin space left parenthesis straight a space plus space straight y right parenthesis dy over dx equals 0 rightwards double arrow space sin space straight a space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus space sin space 2 space left parenthesis straight a space plus space straight y right parenthesis dy over dx equals 0 Hence space proved.

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