Second Derivative Calculator
Compute the second derivative of standard function forms to analyse concavity.
About Second Derivative Calculator
Computes the second derivative f''(x) using the finite difference formula: f''(x) ≈ [f(x+h) − 2f(x) + f(x−h)] / h². The sign of f''(x) determines concavity: positive means concave up, negative means concave down, and zero may indicate an inflection point.
Formula
Second derivative (central difference)
f''(x) ≈ [f(x+h) - 2f(x) + f(x-h)] / h²
- f(x)
- The input function
- x
- Point of evaluation
- h
- Step size (≈1e-5)
Worked example
With the default values below, Second Derivative Calculator returns f''(x) of 5.99999717. Change any input above and every figure here is recalculated in your browser.
Inputs used
- f(x)
- x^4 - 3*x^2 + x
- Evaluate at x
- 1
Results
- f''(x)
- 5.99999717
- f'(x)
- -1.00000000
- Concavity
- Concave up
Step by step
- 1Values used
- 2Second derivative (central difference)
- 3f''(x)= 5.99999717
- 4f'(x)= -1.00000000
- 5Concavity= Concave up
Assumptions and limits
- The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What does the second derivative tell you?
It measures the rate of change of the slope. Positive means the graph curves upward (concave up), negative means it curves downward (concave down).
How do I find inflection points?
Inflection points occur where f''(x) changes sign. Evaluate the second derivative at several points to find where it crosses zero.