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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $left parenthesis 2 straight x plus 3 right parenthesis to the power of straight x minus 5 end exponent$

Solution

Let space straight y equals left parenthesis 2 straight x plus 3 right parenthesis to the power of straight x minus 5 end exponent therefore space log space straight y equals left parenthesis 2 straight x plus 3 right parenthesis to the power of straight x minus 5 end exponent space space space space space space space space space space space rightwards double arrow log space straight y left parenthesis straight x minus 5 right parenthesis. log left parenthesis 2 straight x plus 3 right parenthesis Differentiating space both space sides space straight w. straight r. straight t. space straight x comma space we space get comma 1 over straight y dy over dx equals left parenthesis straight x minus 5 right parenthesis. fraction numerator 2 over denominator 2 straight x plus 3 end fraction plus log left parenthesis 2 straight x plus 3 right parenthesis.1 therefore space dy over dx equals straight y open square brackets fraction numerator 2 left parenthesis straight x minus 5 right parenthesis over denominator 2 straight x plus 3 end fraction plus log left parenthesis 2 straight x plus 3 right parenthesis close square brackets or space dy over dx equals left parenthesis 2 straight x plus 3 right parenthesis to the power of straight x minus 5 end exponent open square brackets fraction numerator 2 left parenthesis straight x minus 5 right parenthesis over denominator 2 straight x plus 3 end fraction plus log left parenthesis 2 straight x plus 3 right parenthesis close square brackets space

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