Question

How do you differentiate `f(x)=ln(x^x)`?

Answer 1

Use logarithm rules to rewrite, then use the Product Rule.

Explanation:

Remember that `log(a^b)=blog(a)`. You can use this identity to rewrite `ln(x^x)` as `xln(x)`.

Now, in order to find `d/(dx)[xln(x)]`, we must use the Product Rule.
The rule dictates that `f'(x)=color(red)(d/(dx)[x])*ln(x)+x*color(blue)(d/(dx)[ln(x)]`
Let's take a moment to determine the derivatives.
`color(red)(d/(dx)[x]=1),color(blue)(d/(dx)[ln(x)]=1/x`

Plug back in.
`f'(x)=color(red)(1)*ln(x)+x*color(blue)(1/x)`
`color(green)(f'(x)=ln(x)+1`