Question

How do you differentiate `y=x^x`?

Answer 1

You can use logarithmic differentiation

Take the natural logarithm of both sides

`lny=lnx^x`

Now using properties of logarithms, rewrite the right hand side

`lny=xlnx`

Differentiate both sides with respect to `x`
Use the product rule on the right side

`1/ydy/dx=lnx+x1/x`

`1/ydy/dx=lnx+1`

Multiply both sides by `y`

`dy/dx=y(lnx+1)`

Now `y=x^x` so we can write

`dy/dx=x^x(lnx+1)`