What is the difference between forward, backward, and central differences?

Forward difference uses f(x₀+h) and f(x₀), backward difference uses f(x₀) and f(x₀-h), and central difference uses f(x₀+h) and f(x₀-h). Central difference provides higher accuracy with O(h²) error compared to O(h) for one-sided differences.

How is the central second derivative calculated?

It is calculated using the formula (f(x₀+h) - 2f(x₀) + f(x₀-h)) / h², providing a second-order approximation of f″(x₀).

Why shouldn't the step size h be extremely small?

While smaller step sizes reduce theoretical truncation error, excessively small values can amplify floating-point round-off errors due to catastrophic cancellation.

What mathematical functions are supported in f(x)?

Standard arithmetic expressions and mathematical functions including sin, cos, exp, ln, sqrt, and power operators are supported.

Is the exact derivative f′(x) required?

No, the exact derivative field is optional. If provided, the calculator computes the absolute error between the numerical approximations and the exact value.

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Numerical Differentiation Calculator (f′(x))

Forward, backward, and central finite differences for f′(x₀) plus the central second derivative, with optional per-method error against the exact derivative.

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2 examples

f = x² at x₀ = 1, h = 0.1 → forward 2.1, backward 1.9, central 2.0

Evaluating x² at 1.1, 1.0, and 0.9 gives 1.21, 1, and 0.81. Forward: (1.21−1)/0.1 = 2.1; backward: (1−0.81)/0.1 = 1.9; central: (1.21−0.81)/0.2 = 2.0 exactly, since central differences are exact for quadratics. The second derivative also lands exactly on 2.

Finite-difference derivatives of f(x) = x^2 at x0 = 1 (h = 0.1) Forward : (f(x0+h) - f(x0))/h = (1.210000 - 1.000000)/0.1 = 2.100000 |error| = 0.100000 Backward: (f(x0) - f(x0-h))/h = (1.000000 - 0.810000)/0.1 = 1.900000 |error| = 0.100000 Central : (f(x0+h) - f(x0-h))/(2h) = (1.210000 - 0.810000)/0.2 = 2.000000 |error| = 0.000000 Second derivative (central): (f(x0+h) - 2f(x0) + f(x0-h))/h^2 = 2.000000

f = sin(x) at x₀ = 0.5, h = 0.01 → central matches cos(0.5) to ~1e⁻⁵

For sin at 0.5 the exact derivative is cos(0.5) = 0.877583. The one-sided differences err by about 4×10⁻³ while the central difference lands within about 1.5×10⁻⁵ — the O(h) vs O(h²) gap in a single table.

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5 options

ContentPaste or type the main input values.2

f(x)textRequiredFunction of x (functions: sin, cos, exp, ln, sqrt, …).Exact Derivative f′(x) (optional)textOptionalExact derivative in x; the tool reports the absolute error of each method. Leave empty to skip.

SettingsAdjust formats, ranges, numbers, and modes.3

Point x₀numberRequiredStep hnumberRequiredDifference step, 0 < h ≤ 1. Central differences: truncation O(h²); too small an h amplifies round-off.Decimal PlacesnumberOptional

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Numerical Differentiation Calculator (f′(x))

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Examples

1. Differentiating a Quadratic Function

2. Approximating Derivatives of Trigonometric Functions

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