Construct a Taylor polynomial about any center or a Maclaurin polynomial about zero. See each derivative, factorial and coefficient, compare the approximation at a point, and graph it against the original function—all locally in your browser.
Choose the function, center and order, then calculate.
A point comparison will appear here when a comparison point is entered.
Derivative and coefficient steps
For each degree k, calculate f⁽ᵏ⁾(a), divide by k!, then multiply by (x−a)ᵏ.
Calculate a polynomial to see every coefficient step.
Function and polynomial graph
Original function Taylor polynomial
Calculate a polynomial to compare it with the original function near the expansion center.
Calculator methodology and limits
Developed and reviewed by
Starlight Robotics calculator team
Last reviewed
Method
Parse the expression → propagate truncated Taylor coefficients through each operation → recover derivative values with f⁽ᵏ⁾(a)=k!cₖ → assemble powers of (x−a). Compact symbolic derivative forms are shown when practical.
Numeric display
Derivative values and coefficients are evaluated as browser floating-point numbers and displayed to about 12 significant digits.
Input limits
One real-valued variable, a 300-character expression, finite center and comparison point up to 1,000,000 in magnitude, and order 0–10.
Privacy
Your function, center, derivatives, results, graph and downloaded text remain on your device.
How to use the Taylor and Maclaurin calculator
Enter a function such as exp(x), sin(x), ln(x), or 1/(1+x^2).
Choose Maclaurin to set a=0, or choose Taylor and enter a custom center.
Select an order from 0 through 10. Order n uses derivatives from f(a) through f⁽ⁿ⁾(a).
Optionally enter a comparison point, then calculate. Review the polynomial, coefficient table, actual error and graph.
A Maclaurin polynomial uses the same formula with a=0. The calculator constructs the finite polynomial Pn; it does not claim that the corresponding infinite series converges to the function at every point.
Center matters
The polynomial matches the function and its first n derivatives at the center. It is generally most useful near that center.
Order matters
A higher order adds derivative information, but it does not guarantee lower error at every point or outside the interval of convergence.
Domain matters
The function and every required derivative must be real and finite at the center. For example, ln(x) cannot be expanded about 0 over the reals.
Error shown
At the optional point, signed error is f(x)−Pₙ(x). This observed difference is not a symbolic remainder bound.
Worked examples
Maclaurin for eˣ
Every derivative of e^x equals e^x, so every derivative at 0 is 1.
P₄(x)=1+x+x²/2+x³/6+x⁴/24.
Maclaurin for sin(x)
Derivative values at 0 repeat 0, 1, 0, −1.
P₇(x)=x−x³/6+x⁵/120−x⁷/5040.
Taylor for ln(x) at 1
Let h=x−1. The signs alternate after the linear term.
P₄(x)=h−h²/2+h³/3−h⁴/4.
Taylor and Maclaurin calculator FAQs
What is the difference between Taylor and Maclaurin series?
A Taylor expansion uses a general center a. A Maclaurin expansion is precisely the Taylor expansion centered at zero.
How does the calculator find each coefficient?
It propagates a truncated power series through the parsed expression using Taylor-mode automatic differentiation. The degree-k coefficient is mathematically the same as f⁽ᵏ⁾(a)/k!; the table shows that derivative value, factorial and coefficient.
Why is the result called a polynomial rather than the full series?
The calculator stops at the selected finite order. An infinite Taylor series also requires a convergence analysis before it can be identified with the original function away from the center.
Does increasing the order always improve accuracy?
No. It commonly improves local accuracy for analytic functions inside their convergence interval, but the outcome depends on the function, center and evaluation point.
Why is my center rejected?
The function or one of the required derivatives is undefined, non-real or infinite there. Examples include ln(x) at 0, 1/x at 0 and abs(x) at its corner.
Are trigonometric inputs in degrees or radians?
Radians. Standard calculus derivative and Taylor-series formulas for trigonometric functions assume radian measure.
Does my expression leave the browser?
No. All parsing, symbolic differentiation, evaluation, graphing and export happen locally on your device.
