Derivative with sample values
[3,−2,0,5] represents 3x³−2x²+5 and differentiates to [9,−4,0].
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Derivative Calculator
Differentiate a polynomial supplied as descending coefficients. A checkable formula accompanies derivative coefficients instead of leaving an unexplained number.
Derivative inputs
Known quantities
Problems suited to Derivative
A coefficient derivative is exact and supports slopes, critical points, Taylor work, and polynomial optimization.
How Derivative works
The power rule multiplies each polynomial coefficient by its exponent and lowers that exponent by one.
Internal zero coefficients must remain in the list to preserve missing powers. The final constant differentiates to zero and disappears. If the Derivative assumptions do not fit, consider numerical slope.
An independent Derivative pass needs Polynomial coefficients, highest power first plus the fixed condition. Judge whether Derivative coefficients has a plausible sign and scale. Test the fixed condition separately; otherwise the cause of a changed Derivative Derivative coefficients remains unclear.
How to challenge the Derivative output
To repeat Derivative, save Polynomial coefficients, highest power first. Translate the output into a sentence about slope, area, curvature, or convergence. This step catches a correct number attached to the wrong calculus quantity, especially when signed accumulation and geometric area are easy to confuse.
For this derivative result, pair the value with its inputs and one short description of what it measures. Naming slope, signed accumulation, geometric area, or convergence removes ambiguity that the number alone cannot resolve.
Label the degree, apply the power rule term by term, and check that the output degree is one lower unless the input is constant. Derivative also leads to evaluate coefficients.
Auditing the Derivative result
Inspect the domain of Polynomial coefficients, highest power first before using Derivative. Keep exact Derivative work separate from the result.
Test Derivative on a constant or linear function. Refine any numerical Derivative step and compare the approximation.
A hand-worked reference for Derivative need not duplicate the original numbers. Select an uncomplicated Polynomial coefficients, highest power first, retain the fixed condition, and estimate Derivative coefficients. The reference gives the original Derivative answer a meaningful scale check.
Questions about Derivative
What happens to a constant?
Its derivative is zero.
Why keep zero coefficients?
They preserve powers.
Is the output symbolic?
It is an exact coefficient list.