



 
Hi, First of all \[\frac{2^x}{x}\] is a fraction so I am going to use the quotient theorem. To accomplish this I need to differentiate the denominator and the numerator. The derivative of the denominator is easy but what about the numerator \[y = 2^{x}?\] When I see a variable in the exponent of an expression I know that I may have to use logarithms. Hence I want to take the natural logarithm of both sides of the expression above to get \[log(y) = log\left(2^x\right) = x \;log(2).\] If I differentiate the left side I get \[\frac{y^{\prime}}{y}.\] and hence $y^{\prime}$ is $y$ times the derivative of the right side of \[log(y) = log\left(2^x\right) = x log(2)\] where $y = 2^{x}.$ I hope this helps,  


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