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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $left parenthesis straight x to the power of straight x right parenthesis to the power of square root of straight x end exponent$

Solution

Let space straight y equals space left parenthesis straight x to the power of straight x right parenthesis to the power of square root of straight x end exponent equals straight x to the power of straight x to the power of begin inline style bevelled 3 over 2 end style end exponent end exponent therefore space log space straight y equals log space straight x to the power of straight x to the power of begin inline style begin display style 3 over 2 end style end style end exponent space space space space end exponent 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 log space straight y equals straight x to the power of begin inline style bevelled 3 over 2 end style end exponent. log space straight x Differentiating space space space both space sides space straight w. straight r. straight t. space straight x comma 1 over straight y dy over dx equals straight x to the power of 3 over 2 end exponent.1 over straight x plus left parenthesis log space straight x right parenthesis.3 over 2 straight x to the power of 1 half end exponent therefore space dy over dx equals straight y open square brackets straight x to the power of 1 half end exponent plus 3 over 2 straight x to the power of 1 half end exponent space log space straight x close square brackets equals left parenthesis straight x to the power of straight x right parenthesis to the power of square root of straight x end exponent open square brackets straight x to the power of 1 half end exponent plus 3 over 2 straight x to the power of 1 half end exponent space log space straight x close square brackets.

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