e^x Calculator — Natural Exponential & Taylor Series
Compute e^x (the natural exponential function) for any value of x, explore its Taylor series expansion, see the derivative and natural log, and visualise the curve. Supports any positive base b^x too.
Mode
e ≈ 2.71828… raised to the power 1
n=0: x^0 / 0!
n=1: x^1 / 1!
n=2: x^2 / 2!
n=3: x^3 / 3!
n=4: x^4 / 4!
n=5: x^5 / 5!
n=6: x^6 / 6!
Partial sum (8 terms)
Exact e^x
- 1
Euler's number e
2.71828183 - 2
Exponent x
1 - 3
e^x
2.718282^(1) = 2.71828183e^x is the unique function equal to its own derivative.
How does this calculator work?
e^x is the natural exponential: e ≈ 2.71828 raised to the power x. Its Taylor series is 1 + x + x²/2! + x³/3! + …, and uniquely, its derivative equals itself. Enter x to get the value, the series partial sum, the graph, and the derivative. Switch to b^x mode for any positive base.
Formula
How this is calculated
The natural exponential function e^x is defined as Euler's number e ≈ 2.71828 raised to the power x. It is the unique function that is its own derivative — d/dx e^x = e^x — which makes it fundamental in differential equations, growth and decay models, probability (the normal and Poisson distributions), and finance (continuous compounding). For any positive base b, b^x = e^(x · ln b), so e^x is the building block for all exponential functions.
The Taylor series e^x = 1 + x + x²/2! + x³/3! + … converges for all real x, and this calculator shows you the partial sum term by term. For small |x| only a few terms are needed; for large |x| many more are required for accuracy. This illustrates why the full series definition must be used for practical computation.
The graph shows e^x (or b^x) in the neighbourhood of your chosen x, revealing the always-positive, ever-increasing exponential growth for positive x and the asymptotic approach to zero for large negative x. The derivative at every point equals the function value itself — a property unique to e^x.
Frequently asked questions
e ≈ 2.71828 is Euler's number, the base of the natural logarithm. It arises naturally as the limit of (1 + 1/n)^n as n → ∞. Its special property is that e^x is its own derivative, making it the natural choice for modelling continuous growth, decay and compound interest.
e^0 = 1 (any non-zero number to the power 0 is 1). e^1 = e ≈ 2.71828. e^(-1) = 1/e ≈ 0.36788.
For x between -2 and 2, about 10 terms give more than 10 significant figures of accuracy. For larger |x|, you need more — at x = 10 you need around 30 terms to converge fully. Modern computers use optimised routines rather than naive Taylor summation for very large arguments.
Also known as
TG we-Calculate Editorial Team. (2026). e^x Calculator — Natural Exponential & Taylor Series [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/e-power-x-calculator
TG we-Calculate Editorial Team. "e^x Calculator — Natural Exponential & Taylor Series." TG we-Calculate. 2026. https://we-calculate.com/calculator/e-power-x-calculator.
TG we-Calculate Editorial Team, "e^x Calculator — Natural Exponential & Taylor Series," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/e-power-x-calculator
@misc{wecalculate_e_power_x_calculator, title = {e^x Calculator — Natural Exponential & Taylor Series}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/e-power-x-calculator}}, year = {2026}, note = {TG we-Calculate} }
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