Polynomial Derivative

d/dx(aₙxⁿ + … + a₁x + a₀) = naₙxⁿ⁻¹ + … + a₁. Power rule.

Inputs

Result

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General calculation reads

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How to use this calculator

  • Enter polynomial coefficients.
  • Optional: evaluate at x.

About this calculator

Power rule: d/dx(xⁿ) = n·xⁿ⁻¹. Constant rule: d/dx(c) = 0. Sum rule: derivative of sum = sum of derivatives. Polynomial differentiation is mechanical: drop the exponent into the coefficient, decrement exponent. Used for finding slopes (tangent lines), velocity from position, optimization (set f' = 0). Source: Wolfram MathWorld - Polynomial Derivative.

Frequently asked

Power rule?+
d/dx(xⁿ) = n·xⁿ⁻¹. Foundation of polynomial differentiation.
Why constant → 0?+
Constants don't change; rate of change is 0.
Use for optimization?+
Set f'(x) = 0 to find critical points. Then check f''(x) for max/min.
Higher derivatives?+
f''(x) = derivative of f'(x). Concavity. Acceleration if f is position.
Non-polynomial?+
Trig, exponential, log have own rules. Chain rule for compositions.

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