What is the second derivative?
The derivative of f′, written f″, measuring how the slope changes.
Second Derivative Calculator
Differentiate twice to get f″(x) for ax²+bx+c or a cubic, and optionally evaluate at x₀.
Enter the polynomial
Choose degree, enter coefficients, and optionally x₀.
About this calculator
How to use Second Derivative Calculator, plus the formulas and logic behind the results.
Tool description
The second derivative measures concavity: f″ > 0 suggests concave up; f″ < 0 suggests concave down.
How to use
- Select quadratic or cubic mode.
- Enter coefficients.
- Optionally enter x₀ to evaluate f″(x₀).
- Select Calculate to view f″(x).
Formulas & reference
Quadratic: f″(x) = 2aCubic: f″(x) = 6ax + 2b
Second derivative tips
- Used in the second derivative test for extrema
- Zeros of f″ are inflection candidates
- Linear and constant functions have f″ = 0
- Sign of f″ describes concavity
Calculation logic
Differentiates twice symbolically and evaluates at x₀ when provided.
For quadratics, f″ is constant—concavity does not change.
Second Derivative Calculator FAQ
Common questions about this calculator and how to use the results.
Quadratic f″?
Always the constant 2a.
Cubic f″?
A linear function 6ax + 2b.
Why evaluate at x₀?
To test concavity or apply the second derivative test at a critical point.
Is f‴ supported?
Not in this tool—stop at f″.
Can I use decimals?
Yes.