Exponentials and Logarithms: The (natural) exponential equals its own derivative and its own antiderivative. Once you accept that the derivative of the natural logarithm is $1/x$ these facts flow from the fact that the exponential is the inverse of the natural logarithm. Both exponentials and logarithms appear frequently in applications.

Here are some exercises reinforcing the unique status of the exponential.
$\frac{\hbox{d} }{\hbox{d} x} 2^x =$ .
$\frac{\hbox{d} }{\hbox{d} x} e^{\sin x} =$ .
$\int x e^{x^2}\hbox{d} x =$ . (As usual ignore the integration constant for indefinite integrals in this home work.)
$\frac{\hbox{d} }{\hbox{d} x} x^{\sqrt{x}} =$ .