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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $cube root of 2 straight x to the power of 4 plus straight x squared minus straight x end root$

Solution

Let space space space space space straight y equals cube root of 2 straight x to the power of 4 plus straight x squared minus straight x end root equals left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis to the power of 1 third end exponent therefore space dy over dx equals 1 third left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis to the power of negative 2 over 3 end exponent. straight d over dx left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis space space space space space space space space space space space space space equals fraction numerator 1 over denominator 3 left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis to the power of bevelled 2 over 3 end exponent end fraction. left parenthesis 8 straight x cubed plus 2 straight x minus 1 right parenthesis equals fraction numerator left parenthesis 8 straight x cubed plus 2 straight x minus 1 right parenthesis over denominator 3 left parenthesis 2 straight x to the power of 4 plus straight x squared minus straight x right parenthesis to the power of bevelled 2 over 3 end exponent end fraction.

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