What is the second derivative?

The second derivative f''(x) is the derivative of the derivative f'(x). It measures the rate of change of the slope — in other words, the curvature or concavity of the original function.

Which functions does this calculator support?

It supports polynomials: terms of the form ax^n where a is any real number and n is any real exponent (including fractions and negatives), plus plain constants.

How do I enter my polynomial?

Use standard notation: 3x^4 + 2x^2 - 5x + 7. Use ^ for exponents. Coefficients must be directly before x with no space (e.g. 3x). Spaces around + and - are fine.

What does the second derivative tell me about concavity?

If f''(x) > 0 at a point, the graph is concave up (cup shape) there. If f''(x) < 0, the graph is concave down (cap shape). A point where f''(x) = 0 and concavity changes is an inflection point.

Can I compute third or higher-order derivatives?

Yes — copy the f''(x) result shown here, paste it into the input, and differentiate again to get f'''(x). For convenience, you can also use our Derivative Calculator to differentiate any intermediate result.

What happens to constants during differentiation?

Any constant term has derivative 0 and is dropped. After the first differentiation a constant in f'(x) will also drop in the second pass, so very low-degree polynomials may simplify to 0.

Second Derivative Calculator

Name: Second Derivative Calculator
Availability: InStock
Author: Nham Vu

Enter a polynomial and instantly get its second derivative f''(x) with step-by-step power rule working.

Enter Polynomial

Use ^ for exponents, e.g. 4x^3 + 2x^2 - 5x + 8

f(x) =

Quick Examples

Enter a polynomial and click Calculate to see f'(x) and f''(x).

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Summary

Enter a polynomial and instantly get its second derivative f''(x) with step-by-step power rule working.

How it works

Type a polynomial expression in the input field, e.g. 4x^3 + 2x^2 - 5x + 8.
Click "Calculate" or press Enter to start.
The power rule d/dx[ax^n] = n·ax^(n−1) is applied to each term to get f'(x).
The power rule is applied a second time to f'(x) to produce f''(x).
All working for both differentiation passes is displayed in a numbered step list.
Copy either result with one click.

Use cases

Verify second derivative homework answers for polynomial functions.
Find where a curve is concave up or concave down using the second derivative sign.
Apply the second derivative test to classify critical points as minima or maxima.
Check acceleration expressions derived from a polynomial position function.
Teach or visualize double differentiation step by step for calculus students.
Quickly confirm d²y/dx² before substituting into a physics or engineering formula.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu