Increasing or Decreasing at a Point Calculator

Enter the coefficients in descending power order and the point to test. The result reports the derivative value, its sign, and the corresponding behavior: increasing for a positive derivative, decreasing for a negative derivative, or stationary when the derivative is zero. It is a focused check for homework, graph analysis, and automated calculus workflows.

What the derivative sign says at one point

A derivative measures the instantaneous rate of change of a function. For a polynomial f, the value f'(a) describes the slope of its graph at x = a. When that value is positive, the tangent line rises from left to right, so the polynomial is classified as increasing at the tested point. When it is negative, the tangent line falls, so the polynomial is classified as decreasing there. A derivative value of zero produces a stationary classification. That last result does not automatically identify a local maximum or minimum: a stationary point may be a turning point, a flat inflection point, or part of a constant polynomial. This calculator deliberately answers the local sign question only. It does not claim behavior over an interval and does not classify critical points. To use it correctly, provide every coefficient in descending power order. For example, coefficients [2, -3, 1] represent 2x² - 3x + 1, while [4, 0, -7] represents 4x² - 7 because the zero preserves the missing x term.

How the polynomial derivative is evaluated

For a polynomial written as a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, differentiation multiplies each coefficient by its exponent and lowers that exponent by one. The constant term disappears. The calculator forms that derivative implicitly and evaluates it with Horner's method, a nested multiplication scheme that avoids constructing powers separately. This is both efficient and numerically simpler than repeatedly calculating x raised to different exponents. Leading zero coefficients are removed before the degree is reported, but zeros inside the coefficient list remain significant placeholders. A one-value list represents a constant polynomial, whose derivative is zero at every finite point. Inputs are limited to 100 finite coefficients, and both coefficients and the point must stay within the published numeric magnitude bound. If intermediate arithmetic exceeds the finite number range, the request returns an input error instead of presenting infinity as a meaningful slope. The reported numeric value is normalized to stable precision so identical inputs produce deterministic JSON results.

Reading the result and avoiding common mistakes

The response separates the computed derivative value, the derivative sign, and the behavior label. This makes the result useful to both a person and an automated workflow: a student can inspect the slope, while code can branch on increasing, decreasing, or stationary without parsing a sentence. The most common input mistake is omitting a zero for a missing power. Coefficients [3, 5] mean 3x + 5, not 3x² + 5; the quadratic must be entered as [3, 0, 5]. Another mistake is treating a stationary result as proof of an extremum. If f'(a) = 0, further analysis is required, such as checking derivative signs on both sides or applying a suitable higher-derivative test. Also remember that this capability evaluates one point, not a whole interval. A polynomial can be increasing at the selected coordinate and decreasing elsewhere. For repeated checks, call the API once per point at $0.002 per request, or use the browser tool for an immediate local calculation. No network lookup, symbolic service, or random process affects the answer.