Derivative with sample values
[3,−2,0,5] represents 3x³−2x²+5 and differentiates to [9,−4,0].
Math calculator
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 this can answer
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.
The calculation treats polynomial coefficients, highest power first according to their labels, not as interchangeable values in derivative.
Reproducing the answer
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.
Mistakes worth catching
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.
Changing only one field helps distinguish a data-entry problem from the intended behavior of derivative.
How the inputs shape the output
A field name is part of the formula. Match the problem's quantities to polynomial coefficients, highest power first, then check that they share the scale assumed by the derivative relationship.
How the result responds
Changing one coefficient affects only its matching derivative power. This is a stronger check than judging the answer only by how many decimal places it shows.
The derivative-at-a-point page accepts a broader function but returns one numerical slope rather than an exact polynomial. The formula panel makes the chosen definition explicit.
Recording the answer
Match the reported precision to polynomial coefficients, highest power first, not to the number of digits the browser can display. Preserve an exact form when it communicates the derivative structure more clearly than a decimal.
The minimum audit trail is short: polynomial coefficients, highest power first, their units, and the formula beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
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.
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.