Compute symbolic derivatives of polynomial functions f(x) = sum of ax^n terms
Enter terms as coeff:power pairs separated by commas. Example: 3:2, -1:1, 5:0 for 3x^2 - x + 5
How it works
A derivative gives the instantaneous rate of change of a function, which geometrically is the slope of the tangent line at a point. This tool differentiates polynomial terms with the power rule and evaluates the result at any x; applying the rule again yields second and higher-order derivatives.
f′(x) = lim h→0 [f(x + h) − f(x)] / h Power rule: d/dx (a·xⁿ) = a·n·x^(n−1) Higher order: f⁽ⁿ⁾(x) is the power rule applied n times in succession
f′(x)first derivative, the slope of f at xf″(x)second derivative, the rate at which that slope changesa, ncoefficient and exponent of a single polynomial termConstant termdifferentiates to zero, which is why f′ loses the trailing constant
- Term by term: 3x⁴ → 12x³, −5x² → −10x, 7x → 7, −2 → 0
- f′(x) = 12x³ − 10x + 7
- f′(2) = 12(8) − 10(2) + 7 = 96 − 20 + 7 = 83
- Differentiate again: f″(x) = 36x² − 10
- f″(2) = 36(4) − 10 = 134
- The sign of the second derivative fixes concavity. Here f″(2) = 134 is positive, so the curve bends upward at that point; a place where f′ = 0 and f″ > 0 is a local minimum.
- The power rule holds for any real exponent, negative and fractional included: d/dx(√x) = d/dx(x^0.5) = 0.5·x^(−0.5).
- Products and quotients cannot be differentiated term by term. Multiply the factors out into a polynomial first, or use the product rule (uv)′ = u′v + uv′.
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Derivative Calculator follows the standard method used in textbooks and technical references for this calculation, and we show the formula or approach on the page itself so you can check the working rather than trust a black box. Found an edge case that breaks it? Use the feedback button in the bottom-right corner.