The formula this page uses
Pₙ(x) = Σ[k = 0…n] f⁽ᵏ⁾(a) · (x − a)ᵏ / k!
Taylor Series Calculator
Build a Taylor or Maclaurin polynomial around any center, choose its degree, inspect every coefficient, and compare the approximation with the original function on an interactive graph.
Derivative at x = 1—Enter a function and calculate.
Show your work Step-by-step method
- The method and numerical checks will appear here.
Calculus
Taylor Series Calculator explained
The short version
- A Taylor polynomial copies a function's value and its first few derivatives at one chosen point.
- Near that point the copy is very good; far away it drifts, and the drift grows fast.
- Centering at a = 0 has a special name, a Maclaurin polynomial, and gives the tidiest formulas.
What each part means
SymbolWhat it means
f(x) — The function being copiedIt must be differentiable n times at the centre.
a — The centreThe point where the copy is made exact. Pick a value where the derivatives are easy.
n — The degreeHow many derivatives get matched. Higher n means a wider window of accuracy.
k! — Factorialk! = k × (k−1) × … × 1, with 0! = 1. It is what keeps the coefficients from exploding.
Show your work: a full example
- The function and centref(x) = eˣ, centred at a = 0, to degree 4
- Every derivative of eˣ is eˣf(0) = f′(0) = f″(0) = f‴(0) = f⁗(0) = 1
- Build the terms1 + x + x²/2! + x³/3! + x⁴/4!
- Work out the factorialsP₄(x) = 1 + x + x²/2 + x³/6 + x⁴/24
- Estimate e^0.51 + 0.5 + 0.125 + 0.0208333 + 0.0026042
- Add them1.6484375
- Compare with the true valuee^0.5 = 1.6487213, so the error is only 0.0002838
A second, different case
- A different case: a function that cannot be centred at 0f(x) = ln x, which is undefined at x = 0, so centre at a = 1 instead
- Derivatives at x = 1f = ln x → 0, f′ = 1/x → 1, f″ = −1/x² → −1, f‴ = 2/x³ → 2
- Build the degree-3 polynomial0 + 1(x−1) + (−1)(x−1)²/2! + 2(x−1)³/3!
- Simplify the factorialsP₃(x) = (x − 1) − (x − 1)²/2 + (x − 1)³/3
- Estimate ln 1.2, so x − 1 = 0.20.2 − 0.04/2 + 0.008/3 = 0.2 − 0.02 + 0.0026667
- Add0.1826667
- Compare with the true valueln 1.2 = 0.1823216, so the error is 0.000345, ten times worse than the eˣ example because 0.2 is a bigger step relative to this function's curvature
Copy-ready example
f(x) = eˣ, centred at a = 0, to degree 4
The function and centre
Questions about the Taylor Series Calculator
What is the difference between a Taylor and a Maclaurin polynomial?
Only the centre. A Maclaurin polynomial is a Taylor polynomial with a = 0, which is why every entry in the table above is written in plain powers of x rather than powers of (x − a).
How do I know how many terms are enough?
Compare consecutive terms. In the e^0.5 example the x⁴ term contributes 0.0026 and the next would contribute about 0.00026, so the error is roughly the size of the first term you dropped. That estimate is the Lagrange remainder in practical form.
Why can ln x not be centred at 0?
Because ln 0 is undefined and every derivative blows up there. Centring at a = 1 works because ln 1 = 0 and all the derivatives are clean whole numbers, which is what the second example uses.
What is this actually for?
Calculators and computers use it. There is no circuitry that knows sin 0.3 directly, so the machine evaluates a polynomial like x − x³/6 + x⁵/120 instead, which needs only multiplication and addition.
A calculator should support the reasoning.
Use numerical results to explore and verify. For a proof or exact symbolic derivation, follow the linked step-by-step solver and check its assumptions.
Derivatives
Estimate local rate of change with centered finite differences and visualize the tangent line.
Integrals
Use adaptive Simpson integration to estimate signed accumulated area over a finite interval.
Limits
Compare values approaching from both sides so one-sided disagreement remains visible.
Taylor models
Approximate a function near a center and compare the polynomial with the original curve.
Numerical-method note: Decimal results are estimates. Discontinuities, oscillation, very large values, and poorly scaled inputs can require symbolic analysis or tighter assumptions.
Learn the mathematics
Understand calculus behind this calculator
A calculator confirms an answer. Working the method yourself is what makes the next problem faster.