Math CentralQuandaries & Queries


Question from Chhavi:

Find f'(x).


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


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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