Question

Using (ln x)' = 1/x, differentiate the function f(x)=x^x

Answer 1

You have the following function:

f(x) = xˣ

In order to differentiate the previous function, you proceed as follow:

apply ln to the function:

ln f(x) = ln (xˣ)

use the property of logarithm ln aⁿ = n(lna)

ln f(x) = x (ln x)

differentiate the previous equation

[ln f(x)]' = ln x + x(1/x) = ln x + 1

[ln f(x)]' = ln x + 1

use [ln f(x)]' = f'(x)/f(x)

f'(x)/f(x) = ln x + 1

f'(x) = f(x)[ln x + 1]

replace f(x) = xˣ

f'(x) = xˣ[ln x + 1]