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Continuity And Differentiability

Question

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$Differentiate space cos space straight x times cos space 2 straight x. cos 3 straight x space space straight w. straight r. straight t. straight x.$

Solution

Let space space space space space space straight y space equals space cos space straight x space times space cos space 2 straight x space cos space 3 space straight x therefore space log space straight y space equals space log space left square bracket cos space straight x space times space cos space 2 straight x space times space cos space 3 straight a right square bracket therefore space log space straight y space equals space log space cos space straight x space plus space log space cos space 2 straight a space plus space log space cos space 3 straight x Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get. space 1 over straight y dy over dx equals fraction numerator negative sin space straight x over denominator cos space straight x end fraction plus fraction numerator negative 2 sin space straight x over denominator cos space 2 straight x end fraction plus fraction numerator negative 3 sin space 3 straight x over denominator cos space 3 straight x end fraction therefore space space space dy over dx equals negative straight y left square bracket tan space straight x plus 2 tan space 2 straight x plus 3 space tan space 3 straight x right square bracket therefore space space space space space space space space space space space equals negative cos space straight x space cos 2 straight x space cos space 3 straight x left square bracket tan space straight x plus 2 tan space 2 straight x plus 3 tan space 3 straight x right square bracket

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