x^x derivative | What is the derivative of x to x?

Good day to all guys. In this article, we will tell you what is the derivative of x to the x. We wish you good lessons in advance.

Derivative of x to x

x to x derivative | x^x derivative

👉 Question : $\frac{d}{dx}x^x=?$

When taking the derivative of x to the x, we first make the x=e^(lnx) transformation. More precisely, it is equal to x=e^(lnx).

👉 $x=e^{lnx}$

So we write e^lnx where we see x. Then our new question becomes as follows;

👉 $\frac{d}{dx}(e^{lnx})^x$

Let's solve our question step by step from here;

👉 $\frac{d}{dx}(e^{x.lnx})$

Now we take the derivative. We write the same and take the product derivative of the exponent.

👉 $(e^{x.lnx}).(x.\frac{1}{x}+lnx.1)$

👉 $x^x.(1+lnx)$

Answer : $x^x.(1+lnx)$