Natural Exponential Function eˣ: Graph, Derivative, Integral and e

The natural exponential function f(x) = e^x is the exponential function whose base is Euler's number e \approx 2.71828. It is the single most important function in calculus, because it is its own derivative: the rate at which e^x grows always equals its current value.

Formula and the number e

Every exponential function b^x has a slope at x = 0 proportional to its height. The base e is the one choice that makes the slope at x = 0 exactly 1, which is why derivatives of e^x come out so clean.

Graph of eˣ

Key properties

• Domain all reals, range (0, \infty): e^x is always positive.
• Always increasing and always concave up, since f'(x) = f''(x) = e^x > 0.
• Horizontal asymptote y = 0: \lim_{x \to -\infty}e^x = 0, while \lim_{x \to \infty}e^x = \infty.
• Inverse of the natural log: e^{\ln x} = x for x > 0 and \ln(e^x) = x for all x. See the natural logarithm.
• Exponent rules: e^{a+b} = e^ae^b, e^{a-b} = \frac{e^a}{e^b}, (e^a)^b = e^{ab}, e^0 = 1.
• Grows faster than any power: \lim_{x \to \infty}\frac{x^n}{e^x} = 0 for every n.
• Power series: e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots, valid for all x.

Derivative of eˣ

With the chain rule:

The exponent is never brought down in front. That is the power rule, which does not apply here.

Integral of eˣ

Worked examples

Example 1. \frac{d}{dx}e^{3x} = 3e^{3x}.

Example 2: chain rule. \frac{d}{dx}e^{x^2} = 2x\,e^{x^2}.

Example 3: product rule. \frac{d}{dx}\left(x e^x\right) = e^x + xe^x = (1 + x)e^x.

Example 4: definite integral. \int_0^{\ln 2}e^x\,dx = e^{\ln 2} - e^0 = 2 - 1 = 1.

Example 5: u-substitution. With u = x^2, \int 2x\,e^{x^2}\,dx = e^{x^2} + C.

Example 6: exponential growth. The solution of y' = ky with y(0) = y_0 is y = y_0e^{kt}. This is why e appears in population growth, radioactive decay and continuous compound interest.

Common mistakes

1. Using the power rule. \frac{d}{dx}e^x \ne xe^{x-1}.
2. Forgetting the chain rule factor. \frac{d}{dx}e^{5x} = 5e^{5x}, not e^{5x}.
3. Integrating e^{kx} without dividing by k. \int e^{4x}\,dx = \frac{1}{4}e^{4x} + C.
4. Treating e^2 as a variable. e^2 is a constant, so its derivative is 0.
5. Writing e^{a+b} = e^a + e^b. Exponents add when you multiply: e^{a+b} = e^ae^b.

How it connects

e^x is the inverse of \ln x and the building block of the hyperbolic functions. Any exponential can be rewritten with base e: b^x = e^{x\ln b}. For a full walkthrough with more practice, read the guide to derivatives of exponential and logarithmic functions, and see Taylor series for its series expansion.