Question

Use the Chain rule to find the derivatives of the following:
$straight f left parenthesis straight t right parenthesis equals open parentheses fraction numerator 2 straight t cubed plus 1 over denominator 3 straight t squared plus 1 end fraction close parentheses squared$

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Solution

straight f left parenthesis straight t right parenthesis equals open parentheses fraction numerator 2 straight t cubed plus 1 over denominator 3 straight t squared plus 1 end fraction close parentheses squared Let space space space space space space space space space space space space space space space straight y equals straight f left parenthesis straight t right parenthesis minus straight u squared comma space where space straight u equals fraction numerator 2 straight t cubed plus 1 over denominator 3 straight t squared plus 1 end fraction. Then space space space space space space space space space dy over du equals 2 straight u and space space space space space space space space space space space du over dt equals fraction numerator left parenthesis 3 straight t squared plus 1 right parenthesis begin display style straight d over dt end style left parenthesis 2 straight t cubed plus 1 right parenthesis minus left parenthesis 2 straight t cubed plus 1 right parenthesis begin display style straight d over dt end style left parenthesis 3 straight t squared plus 1 right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 3 straight t squared plus 1 right parenthesis left parenthesis 6 straight t squared right parenthesis minus left parenthesis 2 straight t cubed plus 1 right parenthesis left parenthesis 6 straight t right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction equals fraction numerator 18 straight t to the power of 4 plus 6 straight t squared minus 12 straight t to the power of 4 minus 6 straight t over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 6 straight t to the power of 4 plus 6 straight t squared minus 6 straight t over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction equals fraction numerator 6 left parenthesis straight t to the power of 4 plus straight t squared minus straight t right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction By space Chain space Rule comma dy over dt equals dy over du du over dt therefore space space space space space space space space space space space space space space space space space space dy over dt equals 2 straight u. fraction numerator 6 left parenthesis straight t to the power of 4 plus straight t squared minus 1 right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction equals 2 open parentheses fraction numerator 2 straight t cubed plus 1 over denominator 3 straight t squared plus 1 end fraction close parentheses. fraction numerator 6 left parenthesis straight t to the power of 4 plus straight t squared minus straight t right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction 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 equals fraction numerator 12 left parenthesis 2 straight t squared plus 1 right parenthesis left parenthesis straight t to the power of 4 plus straight t squared minus straight t right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis cubed end fraction

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Continuity And Differentiability

Use the Chain rule to find the derivatives of the following:
$straight f left parenthesis straight t right parenthesis equals open parentheses fraction numerator 2 straight t cubed plus 1 over denominator 3 straight t squared plus 1 end fraction close parentheses squared$

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