A student is trying to differentiate the function

Here is their working:

According to the rule

$$f(x)=x^n\Rightarrow f^{\prime }(x)=n{x}^{n-1}$$

we have

$$f(x)=e^x\Rightarrow f^{\prime }(x)=x{e}^{x-1}$$

Explain the student's mistake.