How do you solve: d/dx ((1+ 1/x)^x)

Expert Answers

An illustration of the letter 'A' in a speech bubbles

So you need to differentiate function `y(x)=(1+1/x)^x.` The problem is that both base and exponent contain variable `x.` To solve that we first need to take natural logarithm of both sides. ` `


Now we use the following rule for logarithms

`log_a b^c=c cdot log_a b`

Hence we get


Now we can differentiate both sides, but we must keep in mind that `y` is a function of `x` . 



Now we multiply whole expression by `y=(1+1/x)^x.`

`y'=(1+1/x)^x(ln(1+1/x)-1/(1+x))` <--Your solution

Approved by eNotes Editorial Team