Derivative at a Point Calculator
Estimate the first and second derivative of a cubic at any point using the central-difference method, plus the tangent-line equation.
How it works
f'(x) ≈ [f(x+h) − f(x−h)] / (2h) · f''(x) ≈ [f(x+h) − 2f(x) + f(x−h)] / h²
The central-difference formula estimates a slope from function values just to either side of the point, cancelling the leading error term so the estimate is second-order accurate. The symmetric spacing is why it beats a plain forward difference. The second derivative uses the same three points to measure how the slope itself is changing, which tells you the concavity.
Worked example
For f(x) = x² at x = 3, the exact slope is 2x = 6. The central difference [(3.0001)² − (2.9999)²] / 0.0002 returns 6 to full precision, and the second derivative comes out to 2 — matching the constant curvature of a parabola.
Sources: Finite difference — Wikipedia
Frequently asked questions
Why use central difference instead of forward difference?
A forward difference [f(x+h) − f(x)] / h has error proportional to h. The central difference averages a step forward and back, cancelling that term so the error is proportional to h² — far smaller for the same step size.
What does the second derivative tell me?
It measures concavity. A positive f″ means the curve bends upward (like a valley) at that point, negative means it bends downward (like a hill), and zero often marks an inflection point where concavity switches.
Can I differentiate a line or parabola too?
Yes. Set the x³ coefficient a to 0 for a parabola, or set both a and b to 0 for a straight line. A line has a constant slope and zero second derivative.
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This calculator is for educational purposes. Double-check important results before acting on them.