Derivative sign at a point calculator

The sign gives the polynomial's local direction: a positive derivative means it is increasing, a negative derivative means it is decreasing, and a zero derivative marks a stationary point that may need further analysis. Enter coefficients in descending degree order, include zeros for missing powers, and receive a deterministic result suitable for study, checking work, or automated calculations.

Enter the polynomial without ambiguity

Represent the polynomial with its coefficients arranged from the highest power down to the constant term. For example, the array [3, -2, 5, -7] represents 3x³ - 2x² + 5x - 7. Every power needs a position, so x⁴ + 2x - 1 must be entered as [1, 0, 0, 2, -1]. Those zeros are important because they preserve the degree associated with every later coefficient. Supply at least two coefficients, even when the leading coefficient is zero, and use only finite numbers. Then provide the finite numeric point where the derivative should be tested. This explicit representation avoids the uncertainty of parsing typed mathematical notation: there is no question about whether adjacent symbols imply multiplication, how an exponent was written, or where a minus sign belongs. The calculator also reports the degree implied by the array, allowing you to confirm that the polynomial was encoded as intended before relying on the derivative result. Inputs are bounded to keep execution predictable in both the browser and API channels.

Understand how the derivative value is computed

For a polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, the first derivative is naₙxⁿ⁻¹ + (n-1)aₙ₋₁xⁿ⁻² + ... + a₁. The calculator forms those derivative coefficients conceptually and evaluates them at the requested point with Horner's method. Horner evaluation processes the coefficient list in a single bounded pass and avoids separately calculating large powers of x. That makes the procedure efficient and deterministic while returning the same mathematical derivative value. Consider f(x) = 3x³ - 2x² + 5x - 7 at x = 2. Its derivative is 9x² - 4x + 5, so the evaluated value is 36 - 8 + 5 = 33. The returned sign is positive and the behavior is increasing. If arithmetic would overflow the finite numeric range, the request is rejected instead of returning Infinity, because an infinite JSON value would not be a reliable calculator result.

Interpret positive, negative, and zero results

The derivative sign describes the polynomial's instantaneous direction at the selected point. A positive value means the graph rises as x moves through that location, so the reported behavior is increasing. A negative value means the graph falls, and the behavior is decreasing. An exact zero produces stationary, because the first derivative alone does not prove what kind of stationary point is present. It might be a local maximum, a local minimum, or a stationary inflection point; classifying it requires nearby sign checks or a higher-derivative test. The calculator deliberately does not label every zero as a turning point. Keep numeric precision in mind when coefficients or points are decimals: JavaScript numbers use standard floating-point arithmetic, so a theoretically zero expression built from inexact decimals can yield a tiny positive or negative value. The tool reports the sign of the computed value without silently imposing a tolerance. For symbolic certainty, use exact algebra; for numerical work, choose and document a tolerance appropriate to your application.