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 Question from Chhavi: f(x)=2^x/x. Find f'(x).

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,
Penny