Answer
$${e^x}$$
Work Step by Step
$$\eqalign{
& \frac{d}{{dx}}\int_0^x {{e^t}dt} \cr
& {\text{Using The Fundamental Theorem }}\left( {{\text{part 1}}} \right) \cr
& A\left( x \right) = \int_a^x {f\left( t \right)dt{\text{ }} \to {\text{ }}} A'\left( x \right) = \frac{d}{{dx}}\int_a^x {f\left( t \right)dt} = f\left( x \right) \cr
& {\text{In this exercise}} \cr
& f\left( t \right) = {e^t},{\text{ and }}a = 0 \cr
& then \cr
& \frac{d}{{dx}}\int_0^x {{e^t}dt} = {e^{\left( x \right)}} \cr
& \frac{d}{{dx}}\int_0^x {{e^t}dt} = {e^x} \cr} $$
