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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis to the power of 4$

Solution

Let space space space space space straight y equals left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis to the power of 4. therefore dy over dx equals 4 left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis cubed. straight d over dx left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis 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 space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because straight d over dx left parenthesis straight u to the power of straight n right parenthesis equals straight n space straight u to the power of straight n minus 1 end exponent du over dx close square brackets space space space space space space space space space space space space equals 4 left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis cubed. left parenthesis 12 straight x squared minus 10 straight x right parenthesis equals 8 straight x left parenthesis 4 straight x cubed minus 5 straight x squared plus 1 right parenthesis cubed. left parenthesis 6 straight x minus 5 right parenthesis

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