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Continuity And Differentiability

Question

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Differentiate cos x³. sin² (x⁵) w.r.t.x.

Solution

Let space space space space space straight y equals cos space straight x cubed. sin squared left parenthesis straight x to the power of 5 right parenthesis space straight w. straight r. straight t. straight x. therefore space dy over dx equals straight d over dx left square bracket cos space straight x cubed. sin squared left parenthesis straight x to the power of 5 right parenthesis right square bracket space space space space space space space space space space space space space equals cos space straight x cubed. straight d over dx left square bracket. sin squared left parenthesis straight x to the power of 5 right parenthesis right square bracket plus. sin squared left parenthesis straight x to the power of 5 right parenthesis. straight d over dx left square bracket cos space straight x cubed right square bracket space space space space space space space space space space space space space equals cos space straight x cubed.2 sin left parenthesis straight x to the power of 5 right parenthesis. straight d over dx left square bracket sin left parenthesis straight x to the power of 5 right parenthesis right square bracket plus sin squared left parenthesis straight x to the power of 5 right parenthesis. left parenthesis negative sin space straight x cubed right parenthesis. straight d over dx left parenthesis straight x cubed right parenthesis space space space space space space space space space space space space space equals cos space straight x cubed.2 sin left parenthesis straight x to the power of 5 right parenthesis. cos left parenthesis straight x to the power of 5 right parenthesis. straight d over dx left parenthesis straight x to the power of 5 right parenthesis plus sin squared left parenthesis straight x to the power of 5 right parenthesis. left parenthesis negative sin space straight x cubed right parenthesis.3 straight x squared space space space space space space space space space space space space space equals cos space straight x cubed.2 sin space straight x to the power of 5. cos space straight x to the power of 5.5 straight x to the power of 4 minus sin squared space straight x to the power of 5. sin space straight x cubed.3 straight x squared therefore space dy over dx equals 10 straight x to the power of 4 space cos space straight x cubed space sin space straight x to the power of 5 space cos space straight x to the power of 5 minus 3 straight x squared space sin space straight x cubed space sin squared space straight x to the power of 5

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