The Euler constant e and the associated exponential function ex can be defined

in several different ways. The approach adopted here is a geometrical one, and it is motivated by consideration of the derivative of the function

yax (a1)

and the behaviour of the tangent line to the graph of this function when

x/0. To be specific, we take as the defining property of the Euler constant e, that value of a which makes the tangent line to the graph at x/0 have gradient 1. Thus we define e to be the value of a for which


dx [ax]

The function


is then called the exponential function.