Question

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

Answer 1

Take the natural log of both sides, then use implicit differentiation ...

Explanation:

Take natural log of both sides, then use the property of logs :

`lny=ln(lnx)^x=xln(lnx)`

Now, using implicit differentiation, product and chain rules ...

`(1/y)y'=ln(lnx)+(x/lnx)xx1/x=ln(lnx)+1/lnx`

Finally, solve for `y'`

`y'=y[ln(lnx)+1/lnx]=(lnx)^x[ln(lnx)+1/lnx]`

hope that helped

Answer 2

(IGNORE)

Explanation:

Rewrite `f(x)` using the properties of logarithms.

`f(x)=xln(x)`

Now, to find `f'(x)`, use the product rule.

`f'(x)=ln(x)d/dx[x]+xd/dx[ln(x)]`

Find each derivative.

`d/dx[x]=1`

`d/dx[ln(x)]=1/x`

Plug the derivatives back in.

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

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