#### spezialize

##### New member

- Joined
- Sep 27, 2005

- Messages
- 26

- Thread starter spezialize
- Start date

- Joined
- Sep 27, 2005

- Messages
- 26

Take natural logs

\(\displaystyle \ln{y} = \ln{\left(x^x\right)}\)

Log rule: \(\displaystyle \ln{\left(x^n\right)} = n\ln{x}\):

\(\displaystyle \ln{y} = x\ln{x}\)

Differentiate implicitly wrt x, apply the product rule to the RHS:

\(\displaystyle \frac{1}{y} \frac{dy}{dx} = \ln{x} + 1\)

Remember that we had \(\displaystyle y = x^x\) so we get

\(\displaystyle \frac{dy}{dx} = x^x(\ln{x} + 1)\)

- Joined
- Sep 27, 2005

- Messages
- 26

thank you very much for explaining that. Makes sense.