Running a mode 01 Pick the mode: Derivative, Integral, Limit or Taylor. 02 Type `f(x)` and the variable. Syntax is calculator-style (x^2, sin(x)*x, (e^x - 1)/x); the variable defaults to x. 03 Fill the mode's fields (a and b for Integral, the point for Limit, order and expansion point for Taylor) and press Compute . 04 Read the result. Integral adds an error line, Limit prints left and right values, and ⚠ lines flag singularities or non-convergence. 05 Change a field and recompute. The panel keeps no history, so copy the readout first. Verified readouts Same inputs reproduce every row below.
Mode
Input
Readout
Derivative
f(x) = sin(x)*x, var x
cos(x) * x + sin(x)
Integral
f(x) = 1/x, a = 1, b = 2
∫ ≈ 0.6931471806 ; Error estimate: ±7.27e-13
Limit
sin(x)/x, x → 0lim x→0 = 1 ; Left limit ≈ 1, Right limit ≈ 1, Extrapolation error estimate: ±8.88e-16
Limit
(3*x^2 + 2)/(x^2 - 1), x → +∞lim x→+∞ = 3 ; Extrapolation error estimate: ±2.66e-15
Taylor
sin(x), order 5, at 0x -0.166667x^3 +0.008333x^5
The derivative is exact algebra, reordered but identical. The integral of 1/x on [1, 2] is ln 2, printed as 0.6931471806 with its own error. At +∞ there are no left/right lines: there is only one direction to sample.
What Compute is doing Derivative mode differentiates symbolically, so its answer is a formula. The other three are numeric: Integral refines an open quadrature rule where the estimated error is largest, Limit samples each side on shrinking scales and extrapolates, Taylor differentiates repeatedly. That is why Integral and Limit carry an error estimate; Derivative never needs one.
Where the numbers stop
The error line is part of the answer. 1/x on [1, 2] reads 0.6931471806 with error ±7.27e-13 ; digits past that line belong to the method.
A divergence is refused, not fudged. 1/x on [0, 1] fails with "Singularity at the left endpoint x = 0 is not integrable: after 21 halvings the tail is still 1.00× its initial value instead of vanishing" ; ln(x) on [0, 1] also fails instead of returning −1.
A two-sided limit can be nonexistent. abs(x)/x at 0 prints Left limit ≈ -1, Right limit ≈ 1 and the warning that the two-sided limit does not exist.What it does not do: it returns no antiderivative formula and never shows derivation steps. Use Step-by-step for the rule at each step, the graphing calculator to see the function, the matrix panel for determinant, rank or eigen work. Taylor output is truncated and rounded The signature trap sits in Taylor mode. sin(x) to order 5 at 0 reads x -0.166667x^3 +0.008333x^5 : the true coefficients 1/6 and 1/120 are rounded to six decimals, and every coefficient below 1e-12 is dropped without a note. The result is a local approximation, not the series identity — integrate or differentiate it as one, and read it together with the order and point you entered.
Where it fits Checking a derivative by hand Compare sin(x)*x with cos(x) * x + sin(x) , or change the variable to t: t^3 - 2*t returns 3 * t ^ 2 - 2 , catching a derivative taken with respect to the wrong letter.
Integrals and expansions The 1/x row shows the pattern of value plus error; with no closed form, that error line tells you whether the printed digits are meaningful. e^x to order 4 at 0 reads 1 +x +0.5x^2 +0.166667x^3 +0.041667x^4 , enough for a linearisation.
Privacy All four modes compute inside your browser; the expression and every result stay on the device.
References
Wikipedia, Taylor series , en.wikipedia.org （访问日期：2026-10-01）— truncation, coefficients and remainder.
Wikipedia, Integral , en.wikipedia.org （访问日期：2026-10-01）— definite integrals and endpoint integrability.
NIST, Digital Library of Mathematical Functions , §3.5 Quadrature, dlmf.nist.gov （访问日期：2026-10-01）— error estimates in quadrature.
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration , 2nd ed., Academic Press, 1984, sciencedirect.com （访问日期：2026-10-01）— adaptive rules and error control.
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Reviewed by CalcX Editorial Team
Updated 2026-10-01