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Continuity And Differentiability

Question
CBSEENMA12034757
Wired Faculty App

If space straight y equals open vertical bar table row cell straight f left parenthesis straight x right parenthesis end cell cell straight g left parenthesis straight x right parenthesis end cell cell straight h left parenthesis straight x right parenthesis end cell row straight l straight m straight n row straight a straight b straight c end table close vertical bar comma space prove space that space dy over dx equals open vertical bar table row cell straight f apostrophe left parenthesis straight x right parenthesis end cell cell straight g apostrophe left parenthesis straight x right parenthesis end cell cell straight h apostrophe left parenthesis straight x right parenthesis end cell row straight l straight m straight n row straight a straight b straight c end table close vertical bar

Solution
space straight y equals open vertical bar table row cell straight f left parenthesis straight x right parenthesis end cell cell straight g left parenthesis straight x right parenthesis end cell cell straight h left parenthesis straight x right parenthesis end cell row straight l straight m straight n row straight a straight b straight c end table close vertical bar equals straight f left parenthesis straight x right parenthesis open vertical bar table row straight m straight n row straight n straight c end table close vertical bar minus straight g left parenthesis straight x right parenthesis open vertical bar table row straight l straight n row straight a straight c end table close vertical bar plus straight h left parenthesis straight x right parenthesis open vertical bar table row straight l straight m row straight a straight b end table close vertical bar therefore space straight y equals left parenthesis cm minus bn right parenthesis straight f left parenthesis straight x right parenthesis minus left parenthesis cl minus an right parenthesis straight g left parenthesis straight x right parenthesis plus left parenthesis bl minus am right parenthesis straight h left parenthesis straight x right parenthesis Differentiating space both space sides space straight w. straight r. straight t. space straight x comma space we space get comma dy over dx equals left parenthesis cm minus bn right parenthesis straight f apostrophe left parenthesis straight x right parenthesis minus left parenthesis cl minus an right parenthesis straight g apostrophe left parenthesis straight x right parenthesis plus left parenthesis bl minus am right parenthesis straight h apostrophe left parenthesis straight x right parenthesis space space space space space space space space equals open vertical bar table row straight m straight n row straight b straight c end table close vertical bar straight f apostrophe left parenthesis straight x right parenthesis minus open vertical bar table row straight l straight n row straight a straight c end table close vertical bar straight g apostrophe left parenthesis straight x right parenthesis plus open vertical bar table row straight l straight m row straight a straight b end table close vertical bar straight h apostrophe left parenthesis straight x right parenthesis therefore space dy over dx equals open vertical bar table row cell straight f apostrophe left parenthesis straight x right parenthesis end cell cell straight g apostrophe left parenthesis straight x right parenthesis end cell cell straight h apostrophe left parenthesis straight x right parenthesis end cell row straight l straight m straight n row straight a straight b straight c end table close vertical bar

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