The function` f(x) = x^x` has a single horizontal tangent on the interval `x > 0` . Find the equation of this tangent line.
If tangent line is horizontal that means that the slope is 0, hence derivative is equal to 0. So let's differentiate the function:
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.
Now we multiply both sides by `y=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).
To find equation of tangent line we must find function value at point `x=e^-1`.
So equation of tangent line is `y=e^(-e^-1)`.