Second Derivative Calculator
This calculator estimates the second derivative f″(a) of a chosen function at a point and reports whether the curve is concave up or down there.
Function f(x)
Concave up
- 1
f(a)
f(x) = x² at x = 1 = 1 - 2
f(a + h)
h = 0.0001, f(1 + 0.0001) = 1.0002Small step h = 1×10⁻⁴ used for numerical differentiation. - 3
f(a − h)
f(1 − 0.0001) = 0.9998 - 4
f''(a) = (f(a+h) − 2f(a) + f(a−h)) ÷ h²
(1.0002 − 2 × 1 + 0.9998) ÷ 0 = 2
How does this calculator work?
The second derivative f''(a) measures how a function's slope is changing at a point. This tool estimates it numerically as (f(a+h) − 2f(a) + f(a−h)) / h². If f''(a) is positive the curve is concave up at a; if negative it is concave down; near zero suggests an inflection point.
Formula
How this is calculated
Pick a function f(x) from the list and a point a. The calculator uses the central second-difference formula f''(a) ≈ (f(a+h) − 2f(a) + f(a−h)) / h² with a small step h = 1e-4, which approximates the exact second derivative to second order in h.
The sign of f''(a) describes concavity: a positive value means the graph curves upward (concave up, like a cup) near a, a negative value means it curves downward (concave down), and a value essentially zero suggests an inflection point or a locally straight region. The result and concavity label update together.
Inputs are unitless real numbers. The point a must lie in the function's domain — for example ln(x) and √x require a > 0, and 1/x requires a ≠ 0. Because the method is numerical, results for functions with very large curvature or evaluated near singularities can carry small round-off error; the plotted curve near a helps you see the bend visually.
Frequently asked questions
A positive second derivative means the function is concave up (curving upward) at that point, while a negative value means it is concave down. A value near zero often indicates an inflection point.
It uses a finite-difference formula with a tiny step h instead of symbolic differentiation, so there is a small numerical error. For smooth functions away from singularities the estimate is very close to the exact value.
Only points inside the function domain. For ln(x) and √x choose a greater than 0, and for 1/x avoid a = 0; otherwise the function is undefined and no result is shown.
Also known as
TG we-Calculate Editorial Team. (2026). Second Derivative Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/second-derivative-calculator
TG we-Calculate Editorial Team. "Second Derivative Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/second-derivative-calculator.
TG we-Calculate Editorial Team, "Second Derivative Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/second-derivative-calculator
@misc{wecalculate_second_derivative_calculator, title = {Second Derivative Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/second-derivative-calculator}}, year = {2026}, note = {TG we-Calculate} }
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