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