Differentiate or integrate a polynomial — exactly, not numerically, since polynomials
have a closed-form derivative and antiderivative. See the tangent line at a point, or
shade the area under the curve between two bounds.
f(x) = aₙxⁿ + ... + a₁x + a₀
f(x) = aₙxⁿ + ... + a₁x + a₀
How to use this calculator
Choose Derivative or Integral, then fill in the polynomial's coefficients - the result
appears underneath as soon as they're all filled. The point/bounds fields are optional:
leave them blank to just see the derivative or antiderivative formula on its own.
DerivativeComputes f'(x) exactly using the power rule (d/dx(axⁿ) = naxⁿ⁻¹ applied term by
term) - no numerical approximation. Enter a point to also see the slope there and
the tangent line, plotted alongside the curve.
IntegralComputes the antiderivative F(x) + C exactly, term by term. Enter both bounds
to also get the definite integral's value (the Fundamental Theorem of Calculus:
F(upper) − F(lower)), with the area between the curve and the x-axis shaded on the
graph.
Why polynomials onlyPolynomials are the one common function family with a genuinely exact,
closed-form derivative and antiderivative for any degree. Functions like sin(x) or
eˣ need either symbolic differentiation rules or numerical methods - a different
(and much bigger) tool than this one.