Question 375065
i)


{{{y=2^x}}} Start with the given equation.



{{{ln(y)=ln(2^x)}}} Take the natural log of both sides.



{{{ln(y)=x*ln(2)}}} Pull down the exponent.



*[Tex \LARGE \frac{y^{\prime}}{y}=\ln(2)] ... Derive both sides



*[Tex \LARGE y^{\prime}=y\ln(2)] ... Solve for y'



*[Tex \LARGE y^{\prime}=2^x\ln(2)] ... Now plug in {{{y=2^x}}}



Note: the derivative of {{{b^x}}} is {{{b^x*ln(b)}}}


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ii)


{{{y=x^x}}} Start with the given equation.



{{{ln(y)=ln(x^x)}}} Take the natural log of both sides.



{{{ln(y)=x*ln(x)}}} Pull down the exponent.



*[Tex \LARGE \frac{y^{\prime}}{y}=\ln(x)+1] ... Derive both sides



*[Tex \LARGE y^{\prime}=y(\ln(x)+1)] ... Solve for y'



*[Tex \LARGE y^{\prime}=x^x(\ln(x)+1)] ... Now plug in {{{y=x^x}}}