How can we know that x^x is an exponential function or not without drawing the graphic?

Question

In general, exponential function is defined as $a\cdot b^x$, where $a$=coefficient, and $b$= base.

I only knew that the function is exponential function or not, just by drawing the graphic.

But, how can we guess that the function is exponential function or not, for example how can we know that $f(x)=x^x$ is exponential function or not without drawing the graphic?

Thanks

Answer 1

If you differentiate $f(x)=e^{g(x)}=e^{xln(x)}$ you will see, that $g'(x)$ is not a constant. Thus $f(x)=x^x$ is not an exponential function.