Find the derivative of $y=x^x$. Find the derivative of $y=x^x$.
My Attempt:
$$y=x^x$$
Taking $\textrm {ln}$ on both sides, we get:
$$\textrm {ln} y= \textrm {ln} x^x$$
$$\textrm {ln} y = x \textrm {ln} x$$
How do I procees further?

Here is an alternative way to do it, without the use of implicit differentiation:

One can start by writing:
$$y=x^x=(e^{\ln{x}})^x=e^{x\ln{x}}$$
Hence, it now becomes extremely obvious to apply the chain rule to obtain:
$$\frac{dy}{dx}=\frac{d}{dx}(e^{x\ln{x}})=\color{green}{\frac{d}{dx}(x\ln{x})}\cdot e^{x\ln{x}}$$
Now all you need to do is apply the product rule on the green one.
