Is $\exp(x)$ the same as $e^x$? For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of it? 

Another reason we use $\exp(x)$ is when defining it in terms of its power series. At that point, we don't know that $\exp(x)=e^x$ when $x$ is real.
Also, there are general problems using the notation $a^z$ when $z$ is complex. 
$a^z$ is actually a multi-valued function. $e^z$ is thus sometimes ambiguous, so we will, in those cases, prefer $\exp(z)$ to clarify that we are talking about the single-valued function.
