Function library
Exponential Function
f(x) = e^x
The function that is its own derivative — constant relative growth, forever increasing, never touching zero.
The exponential function f(x) = eˣ is the mathematical embodiment of growth proportional to size: money earning compound interest, bacteria doubling in a dish, or a rumor spreading through a crowd. Its defining property is that its rate of change equals its current value — d/dx eˣ = eˣ — which is why it appears whenever a quantity’s growth rate is proportional to the quantity itself. The base e ≈ 2.71828 is the unique number that makes this work.
The graph tells the story at a glance: it passes through (0, 1), creeps almost flat along the x-axis for large negative x (approaching but never reaching 0), then bends upward and climbs ever more steeply for positive x. At x = 1 it equals e ≈ 2.718, at x = 2 it is e² ≈ 7.389, and each unit step multiplies the value by another factor of e. Type e^x into the calculator and compare it with 2^x to see how the base controls the steepness.
Computed properties
Calculated by the graphing calculator's own math engine at build time for e^x.
- Roots in [−10, 10]
- none found
- y-intercept f(0)
- 1.0000
- Local extrema in [−10, 10]
- none found
- Sample values
- f(-2) = 0.1353 · f(-1) = 0.3679 · f(0) = 1 · f(1) = 2.7183 · f(2) = 7.3891
- Derivative f′(1)
- 2.7183
- Integral ∫₀¹ f(x) dx
- 1.7183
Key facts
- Domain: all real numbers; range: y > 0 — eˣ is never zero or negative.
- y-intercept at (0, 1); horizontal asymptote y = 0 as x → −∞.
- Strictly increasing and convex everywhere; no maxima, minima, or inflection points.
- Its own derivative: d/dx eˣ = eˣ; an antiderivative is eˣ itself.
- Exponent laws: e^(a+b) = e^a·e^b, e^(−x) = 1/eˣ, (eˣ)^n = e^(nx).
- e ≈ 2.71828; eˣ outgrows every polynomial as x → ∞.
What the exponential function is
Repeated multiplication is the heart of eˣ: e³ means e·e·e, and the exponent laws e^(a+b) = e^a·e^b extend this to all real powers, including fractions and negatives (e^(−x) = 1/eˣ). The number e itself can be defined as the limit of (1 + 1/n)ⁿ as n grows — the result of compounding 100% interest over infinitely many periods — or as the infinite sum 1 + 1 + 1/2! + 1/3! + ⋯.
What makes e special among all bases is the derivative: d/dx aˣ = aˣ·ln(a), and only for a = e does the ln factor equal 1, leaving the function unchanged by differentiation. Equivalently, eˣ is the unique function satisfying f′ = f with f(0) = 1. That is why e is called the natural base — calculus singles it out.
Shape, asymptote, and growth
For x < 0 the graph hugs the x-axis from above, decaying toward 0 without ever touching it — the horizontal asymptote y = 0 as x → −∞. At x = 0 the curve passes through (0, 1), its y-intercept, and for x > 0 it accelerates upward, convex everywhere (second derivative eˣ > 0) and increasing everywhere (first derivative eˣ > 0). There are no zeros, no turning points, no inflection points: just relentless, smooth, upward-curving growth.
Exponential growth eventually outruns any polynomial: eˣ grows faster than x¹⁰⁰, faster than any fixed power. This is why exponentials model runaway processes — chain reactions, viral spread in its early phase — and also why their inverses, the logarithms, grow so slowly. On a log scale, eˣ becomes the straight line y = x, a handy way to spot exponential data.
Where exponentials appear
Any differential equation of the form dy/dx = ky has the solution y = Ce^(kx): Newton’s law of cooling, radioactive decay (with k < 0), continuously compounded interest, and population growth all follow it. The normal distribution’s bell curve, (1/√(2π))e^(−x²/2), puts an exponential of a quadratic at the center of statistics. In complex analysis, Euler’s formula e^(iθ) = cos θ + i·sin θ fuses exponentials with trigonometry and powers all of AC circuit theory and quantum mechanics.
In computing, exponentials cut both ways: algorithms with exponential time complexity become infeasible as inputs grow, while exponential backoff — waiting 1, 2, 4, 8… seconds between retries — is the standard cure for overloaded servers. The logistic curve, eˣ/(1 + eˣ), tames pure exponential growth with a carrying capacity and models everything from epidemics to neural network activations.
Frequently asked questions
Why is eˣ its own derivative?
By definition e is the unique base for which d/dx aˣ = aˣ·ln(a) has ln(a) = 1. Differentiating eˣ via the limit definition gives eˣ times the limit of (e^h − 1)/h, and e is defined precisely as the number making that limit equal 1. So the slope of the graph at each point equals the function’s height there.
What is e, exactly?
An irrational number approximately 2.71828, definable as the limit of (1 + 1/n)ⁿ as n → ∞ or the sum 1 + 1 + 1/2! + 1/3! + ⋯. Like π, its decimal expansion never repeats. It is the “natural” base because calculus takes its simplest form with it.
Does eˣ ever reach zero?
No. For real x, eˣ > 0 always — the graph approaches the x-axis asymptotically as x → −∞ but never touches it. This follows from eˣ·e^(−x) = e^0 = 1: if eˣ were zero, the product could not be 1.