Graph of the Exponential Function $y = e^x$

$y = e^x$ is the exponential function with base $e = 2.71828\ldots$, Euler's number. Known as the natural exponential function, it has a property no other function shares: differentiating it changes nothing.

Domain and range

The domain is all real numbers and the range is $y > 0$: the value $e^x$ is never zero and never negative.

Unchanged by differentiation

The derivative is the function itself.

It is always positive, so the function increases monotonically, and the second derivative $y'' = e^x$ is positive too, so the curve is convex throughout. The value at a point is precisely the rate of increase at that point, so the larger it gets, the faster it climbs. The only function satisfying $y' = y$ with $y(0) = 1$ is $e^x$, a property that may be taken as its definition.

Asymptotes and limits

As $x \to -\infty$, $e^x \to 0$, so the $x$-axis, the line $y = 0$, is a horizontal asymptote, approached from above since the values stay positive. As $x \to +\infty$ the function diverges to $+\infty$, and faster than any polynomial.

This holds for every $n$, however large.

Notable points

The curve passes through $(0, 1)$, $(1, e) \approx (1, 2.718)$ and $(-1, e^{-1}) \approx (-1, 0.368)$. At $(0, 1)$ the tangent is the line $y = x + 1$, of slope $1$, and near the origin the curve almost coincides with it.

Series representation

The function can be written as an infinite series.

Differentiating each term returns the term before it, which is another way of seeing why the whole is unchanged by differentiation.

Relationships with other functions

Its inverse is the natural logarithm $\ln x$, and the two graphs are reflections of each other in the line $y = x$.

Every exponential is no more than $e^x$ stretched or compressed horizontally. Extended to imaginary arguments it reaches the trigonometric functions, tying the two families together.

Applications

Whenever a quantity changes at a rate proportional to its current size, that is, whenever $y' = ky$, the solution is $y = Ce^{kx}$. With $k > 0$ it describes population growth or compound interest; with $k < 0$ it describes radioactive decay, the cooling of a hot body by Newton's law of cooling, and the discharge of a capacitor. It is the first function to reach for whenever nature grows or decays.