The function` f(x) = x^x`  has a single horizontal tangent on the interval `x > 0` . Find the equation of this tangent line.

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tiburtius | High School Teacher | (Level 2) Educator

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If tangent line is horizontal that means that the slope is 0, hence derivative is equal to 0. So let's differentiate the function:

`y=x^x` 

Now before taking derivative we take natural logarithm of both sides.

`ln y=x ln x`

Now we derivate both sides. Remember that `y` is a function so `ln y` is composite function and thus we must use chain rule for its derivative.

`(y')/y=lnx+1`

Now we multiply both sides by `y=x^x`.

`y'=(lnx+1)/(x^x)` 

We see that `y'` will be equal to 0 if and only if `lnx+1=0` ` `  (fraction is equal to 0 if its numerator is equal to 0).

`lnx=-1`

`x=e^-1`

To find equation of tangent line we must find function value at point `x=e^-1`.

`f(x)=e^(-e^-1)`

So equation of tangent line is `y=e^(-e^-1)`.

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