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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x:

$left parenthesis straight a plus 2 straight x right parenthesis to the power of 7 straight x end exponent$

Solution

Let space straight y equals left parenthesis straight a plus 2 straight x right parenthesis to the power of 7 straight x end exponent therefore space log space straight y equals log left parenthesis straight a plus 2 straight x right parenthesis to the power of 7 straight x end exponent equals 7 left square bracket straight x space log left parenthesis straight a plus 2 straight x right parenthesis right square bracket Differentiating space straight w. straight r. straight t. space straight x comma space we space get 1 over straight y dy over dx equals 7 open square brackets fraction numerator 2 straight x over denominator straight a plus 2 straight x end fraction plus log left parenthesis straight a plus 2 straight x right parenthesis.1 close square brackets therefore space space dy over dx equals 7 straight y open square brackets fraction numerator 2 straight x over denominator straight a plus 2 straight x end fraction plus log left parenthesis straight a plus 2 straight x right parenthesis.1 close square brackets therefore space dy over dx equals 7 left parenthesis straight a plus 2 straight x right parenthesis to the power of 7 straight x end exponent open square brackets fraction numerator 2 straight x over denominator straight a plus 2 straight x end fraction plus log left parenthesis straight a plus 2 straight x right parenthesis.1 close square brackets.

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